There are only a finite number of excluded minors for the class of bicircular matroids
TTHERE ARE ONLY A FINITE NUMBER OF EXCLUDEDMINORS FOR THE CLASS BICIRCULAR MATROIDS
MATT DEVOS ∗ , DARYL FUNK † , AND LUIS GODDYN ∗ Abstract.
We show that the class of bicircular matroids has only afinite number of excluded minors. Key tools used in our proof includerepresentations of matroids by biased graphs and the recently introducedclass of quasi-graphic matroids. We show that if N is an excluded minorof rank at least eight, then N is quasi-graphic. Several small excludedminors are quasi-graphic. Using biased-graphic representations, we findthat N already contains one of these. We also provide an upper bound,in terms of rank, on the number of elements in an excluded minor, sothe result follows. Introduction
Let G = ( V, E ) be a graph, possibly with loops or multiple edges. Thesubgraphs of G that are minimal with respect to having more edges thanvertices are the bicycles of G . The bicircular matroid of G , denoted B ( G ),is the matroid on ground set E whose circuits are the edge sets of bicycles of G . A loopless matroid M is bicircular if there is a graph G for which B ( G )is isomorphic to M .The class of bicircular matroids is minor-closed (after allowing for loops,for instance by including direct sums with rank-0 uniform matroids). Twenty-seven excluded minors for the class of bicircular matroids are known [4]. Weconjecture that this list is complete. Here we prove that the list is finite: Theorem 1.
There are only a finite number of excluded minors for the classof bicircular matroids.
We prove Theorem 1 by showing that there are no excluded minors ofrank greater than seven, then bounding the number of elements, in terms ofrank, that an excluded minor may have.Bicircular matroids form a natural and fundamental class of matroids.When considered within the larger classes containing them, bicircular ma-troids appear as natural and fundamental as the class of graphic matroids.Perhaps the most natural class of matroids in which we should situate our-selves is the class of quasi-graphic matroids, introduced by Geelen, Ger-ards, and Whittle [3]. For a vertex v in a graph G , denote by star( v )the set of edges incident to v whose other end is in V ( G ) − { v } (thusstar( v ) contains no loops). A framework for a matroid M is a graph G Date : February 18, 2021. a r X i v : . [ m a t h . C O ] F e b DEVOS, FUNK, AND GODDYN with E ( G ) = E ( M ) such that (i) the edge set of each component of G has rank in M no larger than the number of its vertices, (ii) for each ver-tex v ∈ V ( G ), cl M ( E ( G − v )) ⊆ E ( G ) − star( v ), and (iii) no circuit of M induces a subgraph of G with more than two components. A matroid is quasi-graphic if it has a framework.Both the cycle matroid M ( G ) and the bicircular matroid B ( G ) of anygraph G are quasi-graphic, sharing the framework G . There will be ingeneral many quasi-graphic matroids sharing G as a framework. The cyclematroid and the bicircular matroid occupy two extremes amongst the quasi-graphic matroids, in the following sense. Let M be a connected quasi-graphicmatroid. Then M has a connected framework G (by Corollary 4.7 of [1]).The cycle matroid M ( G ) of G is the tightest possible quasi-graphic matroidwith framework G . That is, if X ⊆ E ( G ) is a dependant set in a quasi-graphic matroid with framework G , then X is dependant in M ( G ). Thebicircular matroid B ( G ) of G is the freest possible quasi-graphic matroidwith framework G . That is, every independent set in any quasi-graphicmatroid with framework G is independent in B ( G ).The class of graphic matroids is minor-closed. In 1959 Tutte showed thatthe class of graphic matroids is characterised by a list of just five excludedminors (see [5, Theorem 10.3.1]). Our strategy for showing that likewise,the bicircular matroids are characterised by a finite list of excluded minors,is as follows. Let N be an excluded minor for the class of bicircular ma-troids. We show that N cannot be large, neither in rank nor number ofelements. The more challenging part is in bounding the rank. It is not hardto show that N must be vertically 3-connected. Under the assumption that N has rank greater than seven, we find a 3-element set X ⊆ E ( N ) for which N/Z remains vertically 3-connected, for every subset Z of X . Wagner’scharacterisation [6] of the graphs representing a vertically 3-connected bicir-cular matroid of rank at least five enables us to construct a bicircular twin M for N : that is, a bicircular matroid M with E ( M ) = E ( N ) for which M/Z = N/Z for each subset Z ⊆ X . We use the twin M to show that N is quasi-graphic, sharing a framework with M . Using biased-graphic repre-sentations (described in Section 3.1), it is then relatively straightforward toderive the contradiction that N properly contains an excluded minor for theclass of bicircular matroids.We begin with some elementary facts regarding the structure of bicircularmatroids that are not vertically 3-connected, and show that excluded mi-nors are vertically 3-connected (Section 2). In Section 3 we more preciselydescribe representations of bicircular matroids by graphs. Sections 4 and 5are devoted to constructing a bicircular twin for an excluded minor of rankgreater than seven. In Section 6, we use the results of Sections 4 and 5to bound the rank of an excluded minor, at no more than seven. Finally,in Section 7, we bound the number of elements in an excluded minor as afunction of its rank. Together, the main results of Sections 6 and 7 proveTheorem 1. XCLUDED MINORS FOR BICIRCULAR MATROIDS 3 Connectivity and separations in bicircular matroids A separation of a matroid M is a partition ( A, B ) of its ground set; we call A and B the sides of the separation. A k -separation of M is a separation( A, B ) with both | A | , | B | ≥ k and r ( A ) + r ( B ) − r ( E ) < k . A matroid M is k -connected if M has no l -separation with l < k . A vertical k -separation of M is a k -separation ( A, B ) with both r ( A ) , r ( B ) ≥ k . Note that if ( A, B ) isa 2-separation in a connected matroid that is not vertical, then either A or B is a subset of a parallel class.A separation of a graph G is a partition ( A, B ) of its edge set; A and B areits sides . For a subset A of edges of G , we denote by V ( A ) the set of verticesof G with an incident edge in A . A k -separation of G is a separation ( A, B )with | V ( A ) ∩ V ( B ) | = k . A separation ( A, B ) is proper if both V ( A ) − V ( B )and V ( B ) − V ( A ) are non-empty. A graph is k -connected if it has at least k + 1 vertices and has no proper l -separation with l < k . A loop in a graphis an edge with just one end and a loop in a matroid is a circuit of size one;context will prevent any possible confusion.Let M and M (cid:48) be matroids on ground sets E and E (cid:48) , respectively. Assume E ∩ E (cid:48) = { e } , and that e is neither a loop nor coloop in either M or M (cid:48) .The of M and M (cid:48) on basepoint e is the matroid, denoted M ⊕ M (cid:48) ,on ground set ( E ∪ E (cid:48) ) − e whose circuits are the circuits of M \ e and M (cid:48) \ e along with all sets of the form ( C − e ) ∪ ( C (cid:48) − e ), where C is a circuit of M containing e and C (cid:48) is a circuit of M (cid:48) containing e . A connected matroid M has a 2-separation if and only if it can written as a 2-sum M = M ⊕ M ,where each of M and M has at least three elements and is isomorphic to aproper minor of M [5, Theorem 8.3.1]. Thus M has a vertical 2-separationif and only if M can be written as a 2-sum in which each of the summandsis a proper minor of M which has at least three elements and rank at leasttwo.The class of bicircular matroids is not closed under 2-sums. However,there is a special case in which we can find a graph representing a 2-sum oftwo bicircular matroids. Let G and G (cid:48) be graphs with E ( G ) ∩ E ( G (cid:48) ) = { e } ,where e is neither a loop nor coloop in B ( G ) or B ( G (cid:48) ). If e is a loop in G incident with v ∈ V ( G ) and e is a loop in G (cid:48) incident with v (cid:48) ∈ V ( G (cid:48) ), thendefine the loop-sum of G and G (cid:48) on e to be the graph obtained from thedisjoint union of G \ e and G (cid:48) \ e by identifying vertices v and v (cid:48) . Proposition 2.1.
Let G and G (cid:48) be graphs with E ( G ) ∩ E ( G (cid:48) ) = { e } . If H is the loop-sum of G and G (cid:48) on e , then B ( H ) is the 2-sum of B ( G ) and B ( G (cid:48) ) on e .Proof. It is straightforward to check that the circuits of B ( H ) and the cir-cuits of the two sum B ( G ) ⊕ B ( G (cid:48) ) coincide. (cid:3) Equipped with this basic tool, we can now show that an excluded minorfor the class of bicircular matroids is vertically 3-connected. We also usethe notion of the parallel connection of two matroids
M, M (cid:48) on an element
DEVOS, FUNK, AND GODDYN e , denoted P ( M, M (cid:48) ). The 2-sum of M and M (cid:48) on basepoint e is givenby deleting e from the parallel connection of M and M (cid:48) on basepoint e : M ⊕ M (cid:48) = P ( M, M (cid:48) ) \ e . (See [5, Section 7.1] for details). Lemma 2.2.
Let N be an excluded minor for the class of bicircular ma-troids. Then N is vertically 3-connected, and every parallel class has size atmost .Proof. Suppose N contains three elements e, f, g in a common parallel class.Let M = N \ g , and let G be a graph representing M . The circuit { e, f } forms a bicycle in G , so e and f are a pair of loops incident to a commonvertex v . Let H be the graph obtained from G by adding g as a loop incidentto v : H is a bicircular representation of N , a contradiction.Now suppose N contains a series pair e, f . Let M = N/f , and let G be agraph representing M . Let H be the graph obtained from G by subdividing e , and label the two edges of the resulting path e, f . Then H is a bicircularrepresentation of N , a contradiction.Finally, suppose N has a vertical 2-separation, where neither side is aseries class. Then N is a 2-sum N = M ⊕ M , where each of M , M has at least three elements and rank at least two, neither is a circuit, andeach is isomorphic to a proper minor of N . Each of M and M is thereforebicircular. We construct M and M as minors of N as follows. As M hasrank at least two and is not a circuit, and e is neither a loop nor coloopin M , we may choose a circuit C of M containing e as well as a secondelement e (cid:48) . Let M +1 be the matroid obtained from the parallel connection P ( M , M ) of M and M on e by deleting all elements of E ( M ) but C and contracting all elements of C but e and e (cid:48) . Elements e and e (cid:48) are aparallel pair in M +1 , and M = M +1 \ e (cid:48) . Let G +1 be a graph representing M +1 . As { e, e (cid:48) } is a parallel pair of M +1 , e and e (cid:48) are a pair of loops incidentto a common vertex of G +1 . Let G = G +1 \ e (cid:48) . Then G is a representationfor M , and e is a loop in G . Similarly, we may construct a graph G representing M in which e is a loop. Thus by Proposition 2.1 the graph H obtained as the loop-sum of G and G on e is a bicircular representationfor N , a contradiction. (cid:3) We say a matroid M is essentially 3-connected if its cosimplification isvertically 3-connected. Equivalently, M is essentially 3-connected if every2-separation ( A, B ) has the property that one of A or B is a collection ofelements in parallel or in series. Accordingly, we say that a 2-separation( A, B ) is essential if neither A nor B has all of its elements in series or allits elements in parallel. Thus a matroid is essentially 3-connected if it hasno essential 2-separation. Note that the definitions say that a matroid isvertically 3-connected if and only if it is essentially 3-connected and has nonon-trivial series class.Let M be a bicircular matroid represented by the graph G = ( V, E ). Fora subset X ⊆ E , denote by a ( X ) the number of acyclic components of the XCLUDED MINORS FOR BICIRCULAR MATROIDS 5 subgraph G [ X ] induced by the edges in X . Then the rank of a set X ⊆ E is r ( X ) = | V ( X ) | − a ( X ). Given a separation ( A, B ) of the matroid, we callthe components of G [ A ] and G [ B ] the parts of the separation in G . Lemma 2.3.
Let M be a connected bicircular matroid represented by thegraph G . A partition ( A, B ) of E ( G ) is an essential 2-separation of M ifand only if there are just two parts of ( A, B ) that contain bicycles, one A (cid:48) contained in A and the other B (cid:48) contained in B , G consists of A (cid:48) and B (cid:48) together with a path between them, and neither G [ A ] nor G [ B ] is just a setof loops incident to a single vertex.Proof. Let (
A, B ) be an essential 2-separation of M . Since M is connected, G is connected and has no balanced loop. The rank calculation r ( A )+ r ( B ) − r ( M ) = | V ( A ) | − a ( A ) + | V ( B ) | − a ( B ) − | V ( G ) | = 1 implies a ( A ) + a ( B ) = | V ( A ) ∩ V ( B ) | −
1. This implies that each acyclic part consists of a pathlinking a pair of vertices in V ( A ) ∩ V ( B ). Since M is connected and ( A, B )is essential, this implies that there is one part containing a bicycle incidentto each of the starting and ending vertices of this path. Moreover, if there isone acyclic part then there must be at least two acyclic parts. This impliesthat either neither of A nor B consists entirely of a set of loops incident toa single vertex, or there are no acyclic parts; that is, | V ( A ) ∩ V ( B ) | = 1. Inthe latter case, neither A nor B can consist entirely of a set of loops incidentto a single vertex, because such a set is a collection of elements in parallelin M , and ( A, B ) is essential. Finally, if both parts containing bicycles werecontained in one side of the separation, say A , then B would have all of itselements contained in a common series class of M , also contradicting thefact that ( A, B ) is essential.Conversely, suppose (
A, B ) is a partition of E ( M ) that has the formdescribed in the statement of the lemma. Then neither A nor B has all itselements in series or all in parallel, and the connectivity calculation for theseparation yields r ( A ) + r ( B ) − r ( M ) = | V ( A ) ∩ V ( B ) | − a ( A ) − a ( B ) = 1so ( A, B ) is an essential 2-separation of M . (cid:3) We now describe how a bicircular matroid with an essential 2-separationmay become essentially 3-connected after deleting or contracting an element.An edge e in a connected graph is a pendant if after deleting loops it hasan end vertex of degree 1. Its vertex of degree 1 (after deleting loops) is its pendant vertex . A set of edges is a pendant set if they share the same pair ofends and after deleting all but one of them the remaining edge is a pendant.A balloon in a bicircular representation G for a matroid is a maximalset of edges X such that G [ X ] is a subdivision of either a loop, or thegraph consisting of a single edge with a loop incident to one of its ends, and V ( X ) ∩ V ( E ( G ) − X ) = { v } , where if G [ X ] has a vertex of degree 1, thenthis vertex is v ; we call v the balloon’s vertex of attachment . The verticesin V ( X ) − { v } are the internal vertices of the balloon. A line of G is a setof edges Y not contained in a balloon such that G [ Y ] is a maximal pathall of whose internal vertices have degree 2 in G and whose end vertices DEVOS, FUNK, AND GODDYN have degree at least 3. The end vertices of a subgraph induced by a line arethe vertices of attachement of the line; internal vertices of the path are the internal vertices of the line. A balloon or a line is trivial if it consists of justa single edge. For simplicity, we will always refer to a trivial balloon as aloop, and a trivial line as an edge. Throughout the remainder of this paper,the term balloon means “non-trivial balloon” and the term line means “non-trivial line”. We say a graph G has a balloon at x when G has a balloonwhose vertex of attachment is x .The elements in a balloon or a line in a graph G are clearly all containedin a common series class of B ( G ). The converse also holds, unless G is justa bicycle. Proposition 2.4.
Let S be a subset of elements of a bicircular matroid M ,where M is not a circuit and M is connected aside from loops. Let G be agraph representing M . Then S is a non-trivial series class of M if and onlyif S induces a balloon or a line of G .Proof. This follows easily from the fact that a set of edges of G is a circuitof M if and only if it induces a bicycle in G . (cid:3) Proposition 2.5.
Let M be a connected matroid with bicircular represen-tation G , and suppose M has an essential 2-separation ( A, B ) . (i) If there is an element e ∈ A such that M/e is essentially 3-connected,then either • G is a path of length 2 with at least 2 loops on each end vertex,and each of G [ A ] and G [ B ] are disconnected; or • G [ A ] consists of a pendant set of edges (possibly of size 1) anda (possibly empty) set of loops incident to its pendant vertex. (ii) If there is an element e ∈ A such that M \ e is essentially 3-connected,then G [ A ] − e is a balloon.Proof. The statement describes the possibilities given the structure of G de-scribed in Lemma 2.3 under the assumptions that M/e or M \ e is essentially3-connected. (cid:3) We will have occasion to use the following simple fact more than once.
Proposition 2.6.
Let G be a 2-connected graph of minimum degree 3. Let ( A, B ) be a proper 2-separation of G with B minimal. Then there is an edge e ∈ B such that G \ e remains 2-connected.Proof. Let e ∈ B . If G \ e is 2-connected, we are done. If not, then a straight-forward argument uncrossing 2-separations shows that | V ( B ) − V ( A ) | = 1and B has the form of one of the graphs shown in Figure 1. In any case, B contains an edge f such that G \ f is 2-connected. (cid:3) XCLUDED MINORS FOR BICIRCULAR MATROIDS 7
A B A B . . .
A B . . .
Figure 1.
Proper 2-separations (
A, B ) with B minimal.3. Representations
Frame matroids and biased graphs.
The class of frame matroids isan important subclass of quasi-graphic matroids. Cycle matroids of graphsand bicircular matroids are both subclasses of the class of frame matroids.Both have natural representations as frame matroids, which we now describe.A framed matroid is a matroid M having a distinguished basis F with theproperty that every element of M is either parallel to an element of F orspanned by a pair of elements in F . Such a basis is a frame for M . A framematroid is a restriction of a framed matroid. Given a graph G = ( V, E ),the set V of vertices of G may naturally be considered to be the frame for aframed matroid on ground set V ∪ E in which every edge is spanned by itsends. The cycle matroid M ( G ) of G is the restriction to E of this framedmatroid when every cycle of G is a circuit. The bicircular matroid B ( G ) of G is the restriction to E of this framed matroid when every cycle of G isindependant.This follows from work of Zaslavsky, who showed [7] that the class offrame matroids consists precisely of those matroids having a representationas a biased graph ; that is, a graph G together with a distinguished collectionof its cycles B with the property that whenever the intersection of a pair ofcycles in B is a non-trivial path, then the cycle obtained as their symmetricdifference is also in B . This property of B is referred to as the theta property ;a graph formed by the union of two cycles whose intersection is a non-trivial path is called a theta . Cycles in B are said to be balanced , and areotherwise unbalanced . A biased graph ( G, B ) represents a frame matroid M if E ( G ) = E ( M ) and every circuit of M is either a balanced cycle or a bicycleof G that contains no balanced cycle. In this case we write M = F ( G, B ).Let G be a graph. Denote by I ( M ) the collection of independent setsof the matroid M . The fact that I ( M ( G )) ⊆ I ( F ( G, B )) ⊆ I ( B ( G )) holdsfor any collection B of cycles of G satisfying the theta property is easilyseen in terms of biased graph representations. It follows directly from thedefinitions that the cycle matroid of G is the frame matroid F ( G, C ( G )),where C ( G ) is the set of all cycles of G . That is, a graphic matroid is aframe matroid having a representation in which all cycles are balanced. The DEVOS, FUNK, AND GODDYN bicircular matroid B ( G ) of G is frame matroid F ( G, ∅ ). That is, B ( G ) is aframe matroid having a representation in which no cycle is balanced.Equipped with these notions, we may now define the class of bicircularmatroids in terms of biased graph representations, in such a way that theclass is minor-closed and so that minor operations in a graph representationcorrespond precisely to minor operations in the matroid. To that end, let M be a matroid on E , let G = ( V, E ) be a graph, and let B be a subset of theset of loops of G . We say the matroid M is bicircular , represented by thebiased graph ( G, B ), if M = F ( G, B ). That is, M is the matroid on E whosecollection of circuits consists of the balanced loops of G along with the edgesets of bicycles of G that do not contain a balanced loop. In other words,a bicircular matroid is a frame matroid that has a representation ( G, B ) inwhich B contains only loops.When ( G, B ) is a biased graph representing a matroid M , we call a sub-graph induced by the edges of a circuit of M a circuit-subgraph of G . When M is frame and B is empty, the circuit-subgraphs of G are precisely itsbicycles. It will be convenient to have names for the different forms ofcircuit-subgraphs. We call a pair of unbalanced cycles that meet in exactlyone vertex tight handcuffs , and a pair of disjoint unbalanced cycles togetherwith a minimal path linking them loose handcuffs . Thus in general, thecircuit-subgraphs of a biased graph ( G, B ) representing a frame matroid arebalanced cycles, along with thetas and handcuffs that contain no balancedcycle.It is often convenient to speak of paths, cycles, and subgraphs induced bya set of edges as if they consist of just their edge sets. For example, when M is a bicircular matroid represented by the graph G , we may say that atheta subgraph X of G is a circuit of M . By this we mean that E ( X ) is acircuit of M .3.2. Minors of bicircular matroids.
Let ( G, B ) be a biased graph, where B contains only loops. We now define minor operations for ( G, B ) so thatthey agree with the corresponding minor operations in the bicircular ma-troid F ( G, B ). These are specialisations of minor operations for generalbiased graphs, which are defined so that they agree with their correspond-ing operations in a represented frame matroid (see [5, Sec. 6.10]). Theseminor operations for biased graphs may also be applied in a framework fora general quasi-graphic matroid so that they agree with the matroid minor,and we will have occasion to do so in Section 6.3. Though extending theseoperations to the quasi-graphic setting is straightforward, some technicaldetails are required to explain them fully, none of which apply for our ap-plications (see [1] or [3] for these details). We believe the reader will haveno trouble finding the required minors.To delete an element e from ( G, B ): when e is not a balanced loop, define( G, B ) \ e = ( G \ e, B ), while if e ∈ B define ( G, B ) \ e = ( G \ e, B − { e } ). Tocontract an element e of ( G, B ) there are three cases to consider. (1) In the XCLUDED MINORS FOR BICIRCULAR MATROIDS 9 case e is a balanced loop, define ( G, B ) /e = ( G, B ) \ e . (2) When e is an edgewith distinct ends, define ( G, B ) /e = ( G/e, B ). (3) If e is an unbalancedloop, say incident with the vertex u , then let G ◦ e be the graph obtainedfrom G by deleting e and replacing each u - v edge (where v (cid:54) = u ) with a loopincident with v ; if u is now isolated, delete u , and let B ◦ e be the set of loops B ∪ { e (cid:48) : e (cid:48) is a loop incident with u in G } . Define ( G, B ) /e = ( G ◦ e , B ◦ e ).The following fact follows immediately from the definitions. Proposition 3.1.
Let M be a bicircular matroid represented by the biasedgraph ( G, B ) , where B contains only loops. Then for every element e of M , • M \ e = F (( G, B ) \ e ) , and • M/e = F (( G, B ) /e ) . Since in every case ( G, B ) \ e and ( G, B ) /e is a biased graph with no bal-anced cycles other than loops, it is clear that these definitions yield a minorclosed class of frame matroids consisting precisely of the set { B ( G ) : G is agraph } along with all minors of a matroid in this set.When M is a loopless bicircular matroid, there is no reason to use thebiased graph representation ( G, ∅ ), so in this case we may simply refer tothe graph G as a bicircular representation , or simply a representation , for M , and write M = B ( G ).3.3. Characterising representations for a bicircular matroid.
Themain difficulty in proving excluded minor theorems for minor-closed classesof matroids lies in the fact that there often may be many different represen-tations for the same matroid. The class of bicircular matroids is no differentin this respect. However, the variability in graph representations for a givenbicircular matroid is limited and fully understood.Wagner [6] and Coullard, del Greco, & Wagner [2] characterised thegraphs representing a given connected bicircular matroid in terms of thefollowing three operations. Let M be a connected bicircular matroid repre-sented by the graph G .(1) Suppose G has a vertex v such that G has a 1-separation ( H, K )with V ( H ) ∩ V ( K ) = { v } and H − v is acyclic. Let L be a line of H with end vertices v and z . Let e be the edge in L incident to v , andlet w ∈ V ( L ) − v . Let G (cid:48) be the graph obtained from G by redefiningthe incidence of e so that e is incident to w instead of v . Then G (cid:48) isobtained from G by rolling L away from v . The inverse operation isthat of unrolling L to v . Each of G and G (cid:48) are said to be obtainedfrom the other by a rolling operation. We call the vertex v an apex vertex of G .(2) Suppose G has a vertex v such that G has a 1-separation ( H, K )such that V ( H ) ∩ V ( K ) = { v } and there is a vertex u (cid:54) = v that isan end vertex of exactly three lines L , L , L in H , where L hassecond end v , and that L and L share their other end w / ∈ { u, v } in common, and that H − { u, v } is acyclic. Let e be the edge of L
10 DEVOS, FUNK, AND GODDYN ......1 2 3 456 m ≥ m ≥ Figure 2.
Graphs representing rank-3 matroids for whichthe conclusion of Theorem 3.2 does not hold.incident to v , and let e be the edge of L incident to w . Let G (cid:48) bethe graph obtained from G by redefining the incidence of e and e so that e is incident to w instead of v , and e is incident to v insteadof w . Then G (cid:48) is obtained from G by a rotation operation. We call u the rotation vertex of G and G (cid:48) , and call L , L , and L the rotationlines of G ; a rotation line with just one edge is a rotation edge . Wecall the vertex v an apex vertex of G .(3) Let Y be a line (respectively, balloon) of G . Let G (cid:48) be the graphobtained by replacing Y with a line (resp. balloon) with the sameedge set and same vertices (resp. vertex) of attachment. Then G (cid:48) isobtained from G by a replacement operation.We state the main results of [2] and [6] here slightly more precisely thanin the original papers, but in each case the full result stated here is provedin the respective paper, with the exception of the adjective “vertical” inTheorem 3.2 (which is a straightforward extension). Theorem 3.2 is a rep-resentation theorem for 3-connected bicircular matroids of rank at least 5,while Theorem 3.3 does the job for all connected bicircular matroids. Theorem 3.2 ([6], Theorem 5) . Let M be a vertically 3-connected matroidof rank at least 5, and let G and G (cid:48) be bicircular representations for M .Then G is obtained from G (cid:48) by either a sequence of rolling operations or arotation operation. The graphs illustrated in Figures 2 and 3 show that the assumption ofrank at least five in Theorem 3.2 is required [2]. The rank-3 uniformmatroid on six elements is represented by the 3-vertex graph with six edgesshown in Figure 2 with any labelling of its edges. Labels may be swappedon edges as indicated in the pair of three-vertex graphs with m ≥ m ≥
7, each of the graphs at right in Figure 3 has m − , . . . , m ; both graphs providebicircular representations for the same matroid. XCLUDED MINORS FOR BICIRCULAR MATROIDS 11 ... ...12 34 56 m ≥ m ≥ Figure 3.
Graphs representing rank-4 matroids for whichthe conclusion of Theorem 3.2 does not hold.For any connected graph G , let co( G ) be the graph obtained from G bycontracting all but one edge of each balloon and all but one edge of each lineof G . We will see later that co( G ) provides a representation for co( B ( G )).Let G be the family of graphs { G : co( G ) is one of the graphs show in Figure2 or 3 } . Theorem 3.3 ([2], Theorem 4.11) . Let M be a connected matroid, and let G and G (cid:48) be bicircular representations for M . If neither G nor G (cid:48) is in G , then G is obtained from G (cid:48) by either a sequence rolling and replacementoperations, or a sequence of rotation and replacement operations. For convenience, we restate Theorem 3.3 in the form we will frequentlyapplied it:
Lemma 3.4.
Let M be an essentially 3-connected matroid of rank at least5, and let G and G (cid:48) be bicircular representations for M . Then G is obtainedfrom G (cid:48) by either a sequence of rolling and replacement operations or asequence of rotation and replacement operations. We say a connected bicircular matroid M is type 1 if it has a bicircularrepresentation G with a vertex v such that G − v is acyclic, type 2 if M is not type 1 but has a representation as described in paragraph (2) above,and type 3 otherwise. By Theorem 3.3, as long as M has rank at least five,if M has a type 3 representation then M is not type 1 or 2. Accordingly,we call a representation witnessing that M is type 1, 2, or 3, respectively,a type 1, 2, or 3 representation . By Lemma 3.4, if M has rank at leastfive, is essentially 3-connected, and type 3, then M has a unique bicircularrepresentation up to replacement. Theorem 3.2 says that if M has rank atleast five, is type 3 and vertically 3-connected, then M has a unique graphrepresentation.An apex vertex is almost always unique, as we note in the followingstraightforward proposition. Proposition 3.5.
Let G be a type 1 or type 2 representation of a connectedbicircular matroid. Then G has a unique apex vertex unless co( G ) is one of . . .v v u u . . .v v . . .v v . . .v v Figure 4.
Graphs with more than one apex; there may beany number of v - v edges. the graphs shown in Figure 4, in which case v and v are both apex verticesof G . An immediate consequence of Proposition 3.5 is the following.
Proposition 3.6.
Let G be a type 1 or type 2 representation of a vertically3-connected matroid of rank at least 5. Then G has a unique apex vertex. Finding 2-connected type 3 representations
Let M and N be matroids on a common ground set E . We say that M and N are twins relative to a subset X of E , or that M is a twin for N relative to X , if M/e = N/e for every e ∈ X .Let N be an excluded minor for the class of bicircular matroids, andassume N has rank at least 8. Our proof strategy is to construct a bicirculartwin M for N relative to a subset X of their common ground set E . Forreasons that will become clear in the course of the proof, set X will containthree elements, and have the following properties: • for every subset Z ⊆ X , the matroid N/Z = M/Z is vertically3-connected, and • in a graph representing M , the subgraph induced by X has at leastfive vertices.We then use the bicircular twin M to deduce the contradiction that N already contains a smaller excluded minor. To do so, our first step is to findan element e for which N \ e is essentially 3-connected having a 2-connectedbicircular representation.We consider two graphs equal if one may be obtained by just relabellingthe vertices of the other. We say two graph G and H are equal up to rolling,rotation, and replacement , and write “ G = H up to rolling, rotation, andreplacement”, when G may be obtained from H via a sequence of rolling,rotation, or replacement operations, and relabelling of vertices.4.1. Series classes in bicircular representations.
It is easy to see thatif M is a vertically 3-connected bicircular matroid, then for any element e ∈ E ( M ), M \ e has at most two non-trivial series classes, because deletingan edge from any graph representing M may leave at most vertices incidentto exactly two edges, and so may leave at most two lines or balloons. We XCLUDED MINORS FOR BICIRCULAR MATROIDS 13 can show that this is also the case when M is an excluded minor of rank atleast six, at least in the case that M \ e is essentially 3-connected.Let us call a vertex that is incident to exactly two edges deficient . Proposition 4.1.
Let G and H be representations for a connected bicircularmatroid M with V ( G ) = V ( H ) . Let v be a deficient vertex of G . Then v isdeficient in H unless (i) v is in a balloon B of G with deg G ( v ) = 2 , while H [ B ] is a balloonwith deg H ( v ) = 3 ; (ii) G and H are type 1 representations of M , v is the apex vertex in G and H , deg H ( v ) = 3 , and G is obtained from H by rolling a lineaway from v ; or (iii) G and H are type 1 representations of M , v is a vertex of degree 2in a line L of G that has the apex vertex u of G as an end, H isobtained by rolling L away from u , and deg H ( v ) = 3 .Proof. A rotation operation does not change the number of edges incidentto any vertex; neither does a replacement operation applied to a line. Areplacement operation applied to a balloon may change the number of edgesincident to a vertex just as described in (i). A rollup operation may changethe number of edges incident to the apex vertex u of a type 1 representationand a vertex of the line other than the neighbour of u in the line. Thus arollup may cause a vertex to become or cease being deficient just as describedin (ii) and (iii). (cid:3) Let us call a representation of a balloon standard if its vertex of attach-ment has degree 1 and the neighbour of its vertex of attachment has degree3. Call a type 1 graph substandard if every balloon has the apex as its vertexof attachment and every loop is incident to the (same) apex. Let us call agraph representation G for a bicircular matroid standard if • each of its balloons are standard, and in addition, • if G is type 1 then G is obtained from a substandard type 1 graph byrolling exactly one line L away from its apex, where among all lineswith the apex as a vertex of attachment L has the greatest numberof edges.By Theorem 3.3, if G is a non-standard representation, then a standardrepresentation may be obtained from G via rolling and replacement opera-tions. Observe that a standard representation for a matroid M has, amongall graphs representing M , the least number of deficient vertices, providedthe degree of the apex is at least three.We need a couple simple facts before proceeding. Proposition 4.2.
Let N be an excluded minor for the class of bicircularmatroids, with rank at least 6, and let e ∈ E ( N ) . Then neither N \ e nor N/e is a circuit.
Proof.
Suppose to the contrary that N \ e is a circuit. Choose an element s ∈ E ( N ); N \ e/s is a circuit. Let H be a graph representing N/s . Since
N/s \ e is a circuit of rank at least 5, H \ e is a bicycle with at least 5 vertices.Since H has at least five vertices, this implies H has a vertex of degree 2,and so that N/s has a cocircuit of size 2. But this implies that N has acocircuit of size 2, contradicting the fact that N is vertically 3-connected.Now suppose N/e is a circuit. Then either N is a circuit or e is a coloopof N . But circuits are bicircular and N is vertically 3-connected, so this isnot possible. (cid:3) The following fact follows immediately from Propositions 2.4 and 4.2,along with the fact that deleting an element from a vertically 3-connectedmatroid leaves the matroid connected, while contracting an element leavesthe matroid connected, aside from the possibility of creating loops whencontracting an element contained in a non-trivial parallel class.
Proposition 4.3.
Let N be an excluded minor for the class of bicircularmatroids, with rank at least 6. Let e ∈ E ( N ) and let S ⊆ E ( N ) − { e } . Let G be a representation for N \ e or for N/e . Then S is a series class of N \ e or N/e , respectively, if and only if G [ S ] is a line or a balloon. We may now show that deleting an element from an excluded minor leavesat most two non-trivial series classes, subject to the assumption that thedeletion remains essentially 3-connected. In fact, we prove a stronger state-ment.
Lemma 4.4.
Let N be an excluded minor for the class of bicircular ma-troids, with rank at least 6. Let e be an element of N , and assume N \ e isessentially 3-connected. Let G be a standard representation for N \ e . Then G has at most two deficient vertices.Proof. Suppose to the contrary that G has more than two deficient vertices.Each deficient vertex is contained in a line or balloon of G . By Proposi-tion 4.3, the lines and balloons of G correspond precisely to the non-trivialseries classes of N \ e . So N \ e has at least one non-trivial series class. Let S , . . . , S k be the non-trivial series classes of N \ e . Let S = (cid:83) i S i . Eachseries class S i , induces either a line or balloon in G , so G has at least onedeficient vertex in each induced subgraph G [ S i ], i ∈ { , . . . , k } . Claim. k ≤ Proof of Claim.
Suppose k ≥
4. Choose an element s ∈ S , and let H bea graph representing N/s . The matroid N \ e/s is essentially 3-connectedand represented by G/s and by H \ e . Thus by Lemma 3.4, H \ e = G/s upto rolling, rotation, and replacement. Since
N/s has no non-trivial seriesclass, by Proposition 4.3 H has no line nor balloon. Thus deleting e from H leaves at most two deficient vertices. This implies that N \ e/s has at mosttwo non-trivial series classes. But G has at least four lines or balloons H [ S i ], XCLUDED MINORS FOR BICIRCULAR MATROIDS 15 i ∈ { , . . . , } , so G/s has at least three lines or balloons. That is, N \ e/s has at least three non-trivial series classes, a contradiction. (cid:3) Claim. If k = 3, then each non-trivial series class of N \ e has size 2. Proof of Claim.
Suppose to the contrary that | S | ≥
3. Let s ∈ S . Let H be a graph representing N/s . As in the previous paragraph, N \ e/s remainsessentially 3-connected and is represented by both G/s and H \ e . Thus pos-sibly after rolling, rotation, or replacement, H \ e = G/s . As in the previousparagraph, since
N/s has no non-trivial series classes, H has no deficientvertices, and so H \ e has at most two deficient vertices. Thus N \ e/s has atmost two non-trivial series classes. But G/s has three lines or balloons, so N \ e/s has three non-trivial series classes, so this is a contradiction. (cid:3) Claim. N \ e has an element that is not in S . Proof of Claim.
Suppose to the contrary that every element of N \ e is con-tained in a non-trivial series class. By Proposition 4.2, N \ e is not a circuit,so N \ e does not consist of just one non-trivial series class. Because N \ e isconnected, N \ e cannot consist of exactly two non-trivial series classes. So N \ e has exactly three non-trivial series classes. By the previous claim, eachclass contains just two elements. This implies N is a matroid on 7 elements,and so has rank either 6 or 7. But a rank 6 matroid on 7 elements is acircuit, and a rank 7 matroid on 7 elements is free, and both of these arebicircular. (cid:3) So choose an element f ∈ E ( N ) − S . Since N is vertically 3-connectedand has no parallel class of size greater than two, N/f is connected, possiblyaside from a component consisting of a loop. Let H be a graph representing N/f (where, if
N/f has a component consisting of a loop, then H hasa component consisting of a single vertex with a single incident balancedloop). As previously, since N/f has no non-trivial series class, H has nodeficient vertex. But both G/f and H \ e represent N \ e/f so, up to rolling,rotation, and replacement, G/f = H \ e (since by fiat H has a balancedloop incident to an otherwise isolated vertex if necessary, and if so, then f must be an unbalanced loop in G , one of two loops incident to a commonvertex, so that when f is contracted in G , a single balanced loop incident toan otherwise isolated vertex results; apply Theorem 3.3 to the matroid andgraphs obtained by deleting the loop of N \ e/f ).As H has no deficient vertices, H \ e can have at most two deficient ver-tices. Thus N \ e/f has at most two non-trivial series classes. More, since f / ∈ S , contracting f in G cannot reduce the number of deficient vertices.Because G is standard, and f / ∈ S , G/f is standard. Thus among allgraphs representing N \ e/f , G/f has the least number of deficient vertices.Thus, since H \ e has at most two deficient vertices, so does G/f have atmost two deficient vertices. But contracting f in G does not reduce thenumber of deficient vertices of G , since f / ∈ S . Thus G has at most twodeficient vertices. (cid:3) Lemma 4.5 follows immediately from Proposition 2.4, and Lemma 4.4.
Lemma 4.5.
Let N be an excluded minor, of rank at least 6, for the classof bicircular matroids. Suppose N \ e is essentially 3-connected with repre-sentation G . (i) N \ e has at most two non-trivial series classes. (ii) If N \ e has just one non-trivial series class S , then • | S | ≤ , and • if G [ S ] is a line, then | S | ≤ , unless G is type 1 and G [ S ] hasan apex as one end. (iii) If N \ e has two non-trivial series classes, S i ( i ∈ { , } ), then • if G [ S i ] is balloon, then | S i | ≤ , and • if G [ S i ] is a line, then | S i | = 2 . Using Lemmas 4.4 and 4.5 we can now show that for an excluded minor N of rank at least seven, if a graph representing N \ e is type 1 or type 2,then its apex vertex is unique. Proposition 4.6.
Let N be an excluded minor of rank at least seven. As-sume N \ e is essentially 3-connected and that G is a type 1 or type 2 repre-sentation for N \ e . Then G has a unique apex vertex. The apex vertex of G is not an internal vertex of a line or balloon of G .Proof. Suppose to the contrary that G is type 1 or type 2 and does not havea unique apex. Then G has the structure described in Proposition 3.5, andhas at least seven vertices. By Lemma 4.5, N \ e has at most two non-trivialseries classes.Suppose first G is type 1. If N \ e has just one non-trivial series class,then it has size at most four. But this implies G has at most five vertices,a contradiction. If N \ e has two non-trivial series classes, then each has sizetwo or three. But this implies G has at most six vertices, a contradiction.Now suppose G is type 2. Then G is a standard representation of N \ e ,and so has at most two deficient vertices, by Lemma 4.4. But this implies G has at most six vertices, a contradiction.Let v be the apex of G . Clearly, deg G ( v ) must be at least three, so v cannot be an internal vertex of a line or balloon of G . (cid:3) Repairing connectivity.
Let N be an excluded minor of rank at leastseven. The goals of this section is to find an element e ∈ E ( N ) for which N \ e is not only essentially 3-connected, but also represented by a 2-connectedgraph. Lemma 4.7.
Let N be an excluded minor for the class of bicircular ma-troids, of rank at least seven. Assume N has an element e so that N \ e isessentially 3-connected, and let G be a graph representing N \ e . Assume G is not 2-connected, but G has an edge f not contained in a line or balloonsuch that co( G ) \ f is 2-connected. Let H be a graph representing N \ f . Then (i) G \ f = H \ e up to rolling, rotation, and replacement, XCLUDED MINORS FOR BICIRCULAR MATROIDS 17 (ii) each line or balloon of G has an internal vertex that is an end of e in H , and no line or balloon of H is contained in a line or balloonof G , (iii) N \ f is essentially 3-connected and H is 2-connected up to rolling,and, (iv) no balloon of G has more than 3 edges.Proof. Since co( G ) \ f is 2-connected, N \{ e, f } is essentially 3-connected.Both G \ f and H \ e represent N \{ e, f } , so by Lemma 3.4, statement (i)holds: G \ f = H \ e up to rolling, rotation, and replacement.Since G is not 2-connected but N \ e is essentially 3-connected, G has aballoon. By Propositions 4.3 and 4.5 G has at most one other balloon orline. Let S be the set of edges contained in a balloon or line of G . Claim.
No subset R of S is a non-trivial series class of N \ f . Proof of Claim.
Suppose there is such a subset R ⊆ S . By Proposition4.3 H [ R ] is a balloon or line. Write Q = E ( H ) − R . Then e ∈ Q and( Q, R ) is either a proper 1-separation of H where R is a balloon or a proper2-separation of H where R is a line. In either case, ( Q, R ) is a vertical 2-separation of N \ f . The facts that f is not in S , co( G ) \ f is 2-connected, and H \ e = G \ f up to rolling, rotation, and replacement, imply that f ∈ cl N Q .But this implies that ( Q ∪ f, R ) is a vertical 2-separation of N , contrary toLemma 2.2. (cid:3) By the claim neither S , nor any subset of S , is a balloon or line in H . Since H \ e = G \ f up to rolling, rotation, and replacement, this implies statement(ii): if S (cid:48) is a line or balloon of G , then in H edge e has an internal vertexof S (cid:48) as an end, and no line or balloon of H is contained in a line of balloonof G .Now suppose H has a proper 1-separation ( A, B ). Since co( G ) \ f is 2-connected, and G \ f = H \ e up to rolling, rotation, and replacement, up torelabelling the sides of the separation, either(1) V ( A ) ∩ V ( B ) is the vertex of attachment of a balloon S in G , A = S ∪ e , and e has both ends in V G ( S ), or(2) H \ e is a type 1 representation for N \{ e, f } , G has a balloon S , G \ f is type 1 with a line A , and H is obtained from G by unrolling S to its apex, adding e as an edge with at least one end an internalvertex of G [ S ], and rolling A away from the apex.In the first case, since f ∈ cl N ( B ), the 2-separation ( A, B ) of N \ f extendsto a 2-separation ( A, B ∪ f ) of N , a contradiction. Thus the second caseholds; that is, statement (iii) holds.Finally, suppose a balloon S in G has more than three edges, and suppose G [ S ] has vertex of attachment x . Then G [ S ] has at least two deficientvertices. But by Lemma 4.4, G has at most two deficient vertices afterpossibly applying a replacement operation, so | S | = 4. Let A = E ( G ) − S ;( A, S ) is a proper 1-separation of G and a vertical 2-separation of N \ e . Since no subset of S forms a line or balloon in H , in H edge e has both ends in V H ( S ). But this implies that e ∈ cl N ( S ), so the vertical 2-separation ( A, S )of N \ e extends to a vertical 2-separation ( A, S ∪ e ) of N , contradicting thefact that N is vertically 3-connected. (cid:3) We describe outcome (ii) of Lemma 4.7 by saying that the edge e in H repairs a line or balloon of G . Lemma 4.8.
Let N be an excluded minor for the class of bicircular ma-troids, with rank at least seven. If N has an element e such that N \ e isessentially 3-connected and type 3, then N has an element e (cid:48) such that N \ e (cid:48) is essentially 3-connected, type 3, and represented by a 2-connected graph.Proof. Let e ∈ E ( N ), assume N \ e is essentially 3-connected, and let G be atype 3 graph representing N \ e . If G is 2-connected we are done, so assumenot. Since N \ e is essentially 3-connected, G has a balloon S that is a seriesclass of N \ e . By Lemma 4.5, N \ e has at most two non-trivial series classes.If it exists, let S be the second non-trivial series class of N \ e ; otherwise,let S = ∅ . Let s ∈ S and, if S is non-empty, let s ∈ S , and let G (cid:48) bethe graph obtained by contracting all elements in S and S other than s and s Then G (cid:48) is 2-connected with minimum degree at least 3. Note that s is a loop in G (cid:48) . Claim. G (cid:48) has an edge e (cid:48) distinct from s and, if S (cid:54) = ∅ then not in { s , s } ,such that G (cid:48) \ e (cid:48) is 2-connected. Proof of claim. If G (cid:48) is 3-connected, then choose any edge e (cid:48) ∈ E ( G (cid:48) ) otherthan s or s : G (cid:48) \ e (cid:48) is 2-connected, so the claim holds. So assume G (cid:48) is not3-connected. If S is empty, or if there is a proper 2-separation of G (cid:48) withboth s and s in the same side, then let ( A, B ) be a proper 2-separationof G with s in A and, if S is non-empty, both s , s in A , and with B minimal. By Proposition 2.6, there is an edge e (cid:48) ∈ B such that G (cid:48) \ e (cid:48) remains 2-connected.So now assume S is non-empty and every proper 2-separation of G (cid:48) has s and s in different sides. Then for any edge f ∈ E ( G (cid:48) ), if G (cid:48) \ f hasa proper 1-separation, then s , s are in opposite sides of this separation.Thus to check that G (cid:48) \ f remains 2-connected, we just need check that thereremains a pair of internally disjoint paths in G (cid:48) \ f linking the end of s andthe ends of s .Let x be the vertex of G (cid:48) to which s is incident. Let ( A, B ) be a proper2-separation of G (cid:48) , and let V ( A ) ∩ V ( B ) = { a, b } . If x ∈ { a, b } , then moving s to the other side of the separation yields a proper 2-separation with s and s in the same side, contrary to assumption; so neither a nor b is equalto x . If s is a loop, say incident to vertex x , then there must an a - b path,either in A or in B , that avoids { x , x } , else we find that G (cid:48) has a proper2-separation with s , s in the same side. Similarly, if s is an edge withdistinct ends x , x (cid:48) , then there is an a - b path in either A or B that avoids XCLUDED MINORS FOR BICIRCULAR MATROIDS 19 s x x (cid:48) v e v e x s ab w Figure 5 x and at least one of x or x (cid:48) , else we again find that G (cid:48) has a proper2-separation with s , s in the same side.Let W ∈ { A, B } be a side of the 2-separation containing such an a - b path, and let P be an a - b path in W avoiding x (if W contains s ) and atleast one of x or x (cid:48) (if W contains s ). If P has just a single edge e (cid:48) , then G (cid:48) \ e (cid:48) remains 2-connected. So assume P has length greater than 1. Let ae v e v . . . v n − e n b be the simple walk in P from a to b , and assume thatdeleting each edge e i of P leaves G (cid:48) \ e i with a 1-separation. Then each of theend vertices of each edge e i is in a proper 2-separation of G (cid:48) . In particular,there is a proper 2-separation ( Z , W ) whose sides intersect in { v , b } , andthere is a proper 2-separation ( Z , W ) whose sides intersect in { v , w } , forsome vertex w ∈ V ( W ) (see Figure 5). Moreover, s and s are on oppositesides of each of these separations. Vertex v does not have degree 2, so thereis an edge e (cid:48) / ∈ { e , e } incident to v . If e (cid:48) is a loop, we are done: G (cid:48) \ e (cid:48) remains 2-connected. So assume v does not have an incident loop, and let e (cid:48) / ∈ { e , e } be an edge incident to v . If e (cid:48) has other end either b or w , thenwe are done: G (cid:48) \ e (cid:48) remains 2-connected. So assume this does not occur;let u be the other end of e (cid:48) . Our assumptions now imply that b (cid:54) = w , sinceotherwise we find a proper 2-separation of G (cid:48) (whose sides meet in { v , b } )with s and s in the same same. Let Z ∈ { A, B } − W be the other side ofour proper 2-separation ( A, B ). Then Z ⊂ Z ⊂ Z , W ⊃ W ⊃ W , and e (cid:48) ∈ Z ∩ W . Let X be the graph obtained from G (cid:48) by replacing each of Z and W with a single edge. That is, X consists of W ∩ Z along with an a - b edge (replacing Z ) and a v - w edge (replacing W ). There is a b - w path in X , and it cannot be the case that every b - w path in X contains e (cid:48) , for then v would be cut vertex of G (cid:48) , contradicting the fact that G (cid:48) is 2-connected.Hence there is a b - w path in X that does not contain e (cid:48) . Thus in G (cid:48) \ e (cid:48) thereis a pair of internally disjoint paths linking x and the ends of s . In otherwords, G (cid:48) \ e (cid:48) remains 2-connected. (cid:3) Apply the preceding claim: let e (cid:48) ∈ E ( G (cid:48) ) − { s , s } (respectively, let e (cid:48) ∈ E ( G (cid:48) ) − { s } if S is empty) be an edge such that G (cid:48) \ e (cid:48) remains 2-connected, and let H be a graph representing N \ e (cid:48) . By Lemma 4.7, H is2-connected up to a rolling operation. Thus N \ e (cid:48) is essentially 3-connected.If H is type 3 we are done. So assume H is a type 1 or type 2 graph representation for N \ e (cid:48) . Wemay assume H is a substandard representation for N \ e (cid:48) (that is, if H istype 1, then H does not have a balloon that may be rolled up to its apex),so H is 2-connected. By Lemma 4.6 H has a unique apex vertex v . Observethat N \{ e, e (cid:48) } is essentially 3-connected. Since H \ e = G \ e (cid:48) up to rolling,rotation, and replacement, G \ e (cid:48) is type 1 or 2 with apex v . By Proposition4.6, v is not an internal vertex of a balloon or line in H . Claim.
Other than its balloon S , G has no other balloon nor a line. Proof of Claim.
Suppose that G has a second line or balloon S . By Lemma4.7, e repairs both of S and S in H . Then not both S and S are balloonswith v as their vertex of attachment (if so, H would have v as a cut-vertex,but H is 2-connected). Since v is not an internal vertex of a balloon or aline, this implies that H is type 3, a contradiction. (cid:3) Claim. H has an edge f (cid:54) = e , incident to v , such that H \ f remains 2-connected. Proof of Claim.
Since H \ e = G \ e (cid:48) up to rolling, rotation, and replacement, V G ( S ) ∪ { v } = V H ( S ) ∪ { v } ; let us denote this set by V S , and by V − S theset of vertices obtained by removing the vertex of attachment of S from V S . Now because H is type 1 or 2 and 2-connected, both ends of e in H are in V S . By Lemma 4.7, | S | ≤
3, so | V − S | ≤ G is type 3, but G \ e (cid:48) is not, v is not an end of e (cid:48) in G . Thus the factthat G \ e (cid:48) = H \ e up to rolling, rotation, and replacement, implies that theonly vertices of H that are possibly of degree less than three are the ends of e (cid:48) in G . Let us denote by T the tree ( G − V − S ) \ e (cid:48) = H − V − S . Since T has atleast four vertices, of which at most two are of degree 2 in H , connectivitynow implies that there is either a leaf of T with at least two incident edgeswhose other end is v , or there is a non-leaf of T with an incident edge whoseother end is v , or the vertex of attachment of S is a leaf of T and so mustbe an end of an edge whose other end is v . Choose such an edge f : H \ f remains 2-connected. (cid:3) Let K be a graph representing N \ f . Because H \ f is 2-connected, N \{ e (cid:48) , f } is essentially 3-connected, so N \ f is essentially 3-connected. By our choiceof f , G \ f remains type 3. Therefore, up to rolling, rotation, and replace-ment, • K \ e = G \ f , so K is type 3 and e (cid:48) has the same ends in K as in G ,and • K \ e (cid:48) = H \ f ; the proof of statement (iii) of Lemma 4.7 shows that e must have the same ends in K as in H .Thus K is a 2-connected type 3 representation for N \ f . (cid:3) XCLUDED MINORS FOR BICIRCULAR MATROIDS 21 Finding X Given an excluded minor N , of rank at least eight, the goal of this sectionis to find a triple of elements X = { a, b, c } ⊆ E ( N ) and a bicircular twin M for N relative to X , such that • X spans at least five vertices in each representation for M , • for each subset Z of X , N/Z is vertically 3-connected.5.1.
Contractable edges.
We use the Edmonds-Gallai decomposition ofgraphs without a perfect matching. For any subset U of vertices of a graph G , denote by N ( U ) the set of neighbours of U , N ( U ) = { y ∈ V ( G ) − U :there is an edge xy ∈ E ( G ) } . A graph is hypomatchable if for every vertex v ∈ V ( G ), G − v has a perfect matching. Theorem 5.1 (Edmonds-Gallai) . Let G be a graph and let A ⊆ V ( G ) bethe set of vertices v such that G has a maximum matching that does notcover v . Set B = N ( A ) and C = V ( G ) − ( A ∪ B ) . Then (i) Every odd component H of G − B is hypomatchable and satisfies V ( H ) ⊆ A . (ii) Every even component H of G − B has a perfect matching and sat-isfies V ( H ) ⊆ C . (iii) For every set U ⊆ B , the set N ( U ) contains vertices in more than | U | odd components of G − B . Call the set B defined in the Edmonds-Gallai Theorem the barrier set of G . An edge e in a 2-connected graph G is contractable if G/e remains2-connected.
Proposition 5.2.
Let G = ( V, E ) be a 2-connected graph on at least 4vertices, and let S ⊆ E be the set of contractible edges of G . Then ( V, S ) is2-connected.Proof. If G is 3-connected, then every edge is in S and the result holds.So assume G has a proper 2-separation. We proceed by induction on thenumber of vertices of G . The result clearly holds when | V ( G ) | = 4, soassume | V ( G ) | > | V ( G ) | vertices. Let ( A, B ) be a proper 2-separation of G with B minimal. Then B ⊆ S . Let G (cid:48) = ( V (cid:48) , E (cid:48) ) be the graph obtained from G by replacing B with a single edge linking the two vertices in V ( A ) ∩ V ( B ). Let S (cid:48) be theset of contractable edges of G (cid:48) . By induction, ( V (cid:48) , S (cid:48) ) is 2-connected. Let x , y be the vertices in V ( A ) ∩ V ( B ). Let R be the set of edges of G thatare either loops incident to x or y or have both x or y as ends. While noedge in R is contractable in G , every edge in R is contractable in G (cid:48) . Byminimality, no edge in R is in B . Thus S = ( S (cid:48) − R ) ∪ B , and so ( V, S ) is2-connected. (cid:3)
We denote by si( G ) the graph obtained from a graph G by replacing everyset of parallel edges with a single edge and removing all loops. Denote by K (cid:48) ,n the graph obtained by adding an edge linking the two degree n verticesof K ,n . Proposition 5.3.
Let G be a 2-connected graph on at least 6 vertices, andlet S be the set of contractable edges of G . Then G [ S ] has a matching ofsize 3 unless si( G ) is K ,n or K (cid:48) ,n .Proof. Suppose G [ S ] does not have a matching of size 3. Then its barrierset B has size at most 2. If | B | = 1, then G [ S ] has the single vertex in B as a cut vertex, contradicting Proposition 5.2. So | B | = 2, and si( G [ S ]) isisomorphic to K ,n , where n ≥
4. Let e (cid:48) be the edge of G linking the twovertices of degree n + 1 of si( G [ S ]), if it exists. It is easy to see that if G had an edge e , other than e (cid:48) , that is not in S , then e would nevertheless becontractable a contradiction. Thus si( G ) is K ,n or K (cid:48) ,n . (cid:3) Proposition 5.4.
Let G be a 2-connected graph and let X be a matchingin G . If G/X is 2-connected, then
G/Z is 2-connected for every Z ⊆ X .Proof. Let Z ⊆ X , and let u, v ∈ V ( G/Z ). It is straightforward to see thatsince there is a pair of internally disjoint u - v paths in G/X , there is also apair of internally joint u - v paths in G/Z . (cid:3) Proposition 5.5.
Let G be a 2-connected graph and let X be a matchingof three edges in G [ S ] , where S is the set of contractable edges of G . Then G/Z is 2-connected for every subset Z ⊆ X .Proof. By Proposition 5.4, we just need to show that
G/X is 2-connected.Let X = { a, b, c } ⊆ E ( G ). Suppose for a contradiction that G/X hasa proper 1-separation. Because a ∈ S , G/a does not have a proper 1-separation. So there must be a k -separation ( A, B ) of
G/a , where k ∈ { , } ,such that G [ { b, c } ] is a path containing V ( A ) ∩ V ( B ). This implies that b and c share an end in G/a . Since b and c do not share an end in G , thisimplies a shares an end with each of b and c . But this is not the case, as X is a matching in G . (cid:3) Staying type 3.
When constructing our twin for an excluded minor,relative to a set X , we would like not only to remain vertically 3-connectedeach time a subset of X is contracted, but also remain of the same type witheach contraction. In this subsection, we consider type 3 representations. Inthe next, we consider types 1 and 2. Proposition 5.6.
Let X be a matching of three edges in a 2-connected type3 graph G with at least seven vertices, such that G/Z is 2-connected forevery subset Z of X . Suppose that there is a subset Y of X such that G/Y is no longer type 3. Then there is a set X (cid:48) of three edges spanning at leastfive vertices for which G/Z is 2-connected and type 3, for every subset Z of X (cid:48) .Proof. Let X = { a, b, c } . After possibly permuting the labels of edges a , b , c , one of the following holds. XCLUDED MINORS FOR BICIRCULAR MATROIDS 23 (1) There is a vertex x such that G − x is type 1 (resp. type 2) withapex vertex v , and a has ends v, x .(2) There is a vertex x such that ( G − x ) \ b is type 1 with apex v , a hasends v, x , and b is in a parallel pair.(3) G \ a is type 2 with apex v and rotation vertex u , a is incident to v and is in a parallel pair { a, f } , and f is in the line of G \ a whoseends are v and u .(4) G has a vertex v such that G − v is acyclic aside from exactly twoloops that are not incident to the same vertex, and a has ends v and x , where x is incident to one of the loops of G − v .(5) G has a vertex v such that G − v is acyclic aside from exactly one loopand one pair of parallel edges, a has ends v, x , where x is incidentto the loop of G − v , and b is one of the edges in the parallel pair.(6) Edge a is in a parallel pair; G \ a is type 1 with apex v and has noloop that is not incident to v .Let S be the set of contractible edges of G . In each case, we find a triple { a (cid:48) , b (cid:48) , c (cid:48) } of edges as desired in G [ S ].In cases (1) and (2) there is a cycle C in G − v (resp. in G − { u, v } , where u is the rotation vertex when G − x is type 2). If | C | ≥
5, we may choose a (cid:48) , b (cid:48) , c (cid:48) as required from the edges of C . If | C | ≤
4, then because G has atleast 7 vertices, we can choose a (cid:48) from among the edges of C , b (cid:48) from amongthose edges incident to a vertex in C that do not have v as an end, and c (cid:48) from the set of edges incident to v that are not incident to a vertex in C noran end of b . Our rank and connectivity assumptions guarantee that suchedges exist.In the remaining cases, all edges in G − v are in S , and G − v consists ofa tree, after possibly removing some edge or edges that we wish to preservein G/X (cid:48) in order to ensure that
G/X (cid:48) remains type 3; namely, an edge ina parallel pair incident to the vertex of rotation when G \ a is type 2, loopsnot incident to v , or an edge from a parallel pair. It is easy to see that ineach case we can find a triple a (cid:48) , b (cid:48) , c (cid:48) in this tree, as desired. (cid:3) Lemma 5.7.
Let N be an excluded minor for the class of bicircular ma-troids, of rank at least seven. Let e ∈ E ( N ) and assume that G is a 2-connected type 3 representation for N \ e . Then there are edges a, b, c ∈ E ( G ) spanning at least five vertices such that for every subset Z ⊆ { a, b, c } , G/Z remains 2-connected and type 3.Proof.
Let S be the set of contractable edges of G . If G [ S ] has a matching { a, b, c } of size 3, then by Proposition 5.5, G/Z is 2-connected for every Z ⊆ { a, b, c } . If G/Z remains type 3 for every Z ⊆ { a, b, c } , we are done.Otherwise, by Proposition 5.6, there are three edges a (cid:48) , b (cid:48) , c (cid:48) as desired, andwe are done.So assume that G [ S ] does not have a matching of size 3. By Lemma 5.3,si( G ) is K ,n or K (cid:48) ,n . Let u , u be the two vertices of degree n or n + 1 insi( G ), and let W = { x , . . . , x n } be the set of vertices of degree 2 in si( G ). Our rank assumption implies that n ≥
7. By Lemma 4.5, at most two of x , . . . , x n have degree 2 in G , so there are at least five vertices in W ofdegree at least three. Because G is type 3, up to relabelling u and u , oneof the following holds: • there are at least two loops in G [ W ], • there is just one loop incident to a vertex in W , but there is a loopincident to each of u and u , • there is just one loop incident to a vertex in W , a loop incident to u , and a vertex in W with at least two edges linking it to u , • there is just one loop in G , incident to a vertex in W , there is avertex in W with at least two edges linking it with u , and a vertexin W with at least two edges linking it with u , or • G has no loops, and – there are either at least two vertices in W sending two edges to u or there is a vertex in W sending at least three edges to u ,and – there are either at least two vertices in W sending two edges to u or there is a vertex in W sending at least three edges to u .It is straightforward to check that in any case a triple of edges exists asrequired. (cid:3) Observe that the property of being type 1 or type 2 (equivalently, theproperty of not being type 3) is closed under minors: that is, if G is a type1 graph, then every minor of G is type 1, while if G is a type 2 graph, thenevery minor of G is either type 2 or type 1. This easy observation will beused more than once. Proposition 5.8.
Let M (cid:48) be a connected minor of a connected bicircularmatroid M , with r ( M (cid:48) ) ≥ . If M (cid:48) is type 3 then M is type 3.Proof. Let G (cid:48) be a graph representing M (cid:48) and let G be a graph representing M . Let C, D ⊆ E ( M ) such that M/C \ D = M (cid:48) . By Theorem 3.3, G/C \ D = G (cid:48) , up to rolling, rotation, and replacement. Since G (cid:48) is type 3, G must alsobe type 3. (cid:3) Lemma 5.9.
Let N be an excluded minor for the class of bicircular ma-troids, of rank at least eight. Let e ∈ E ( N ) and assume that G is a 2-connected type 3 representation for N \ e . Let Z be a set of at most threeedges for which G/Z remains 2-connected and type 3, and let H be a graphrepresenting N/Z . Then H is 2-connected and type 3, and N/Z is vertically3-connected.Proof.
Since
G/Z is 2-connected, N \ e/Z is essentially 3-connected. Thematroid N \ e/Z is represented by both G/Z and H \ e , and since N has rankat least eight, G/Z and H \ e have at least five vertices. Thus by Lemma 3.4, G/Z = H \ e up to rolling, rotation, and replacement. Since G/Z is type 3,rolling and rotation are irrelevant:
G/Z = H \ e up to replacement. Since XCLUDED MINORS FOR BICIRCULAR MATROIDS 25
G/Z is a type 3 minor (up to replacement) of H , by Proposition 5.8, H istype 3.Now suppose H has a proper 1-separation ( A, B ), say with V ( A ) ∩ V ( B ) = { x } and e ∈ A . Then neither A nor B is a parallel class of N/Z . Since
N/Z has no non-trivial series class, (
A, B ) is an essential 2-separation of
N/Z .Since
N/Z \ e is essentially 3-connected, by Proposition 2.5, A − e is a balloonin H \ e . But H \ e = G/Z up to replacement, and
G/Z is 2-connected, sothis is impossible. Thus H is 2-connected. Since N/Z is represented by a2-connected graph,
N/Z is essentially 3-connected. But
N/Z has no non-trivial series class, so
N/Z is vertically 3-connected. (cid:3)
Our main objective for this section now follows from Lemmas 5.7 and 5.9:
Lemma 5.10.
Let N be an excluded minor for the class of bicircular ma-troids, of rank at least eight, and let e be an element such that N \ e isessentially 3-connected and represented by a 2-connected type 3 graph G .Then there is a bicircular twin M for N relative to a set X of size three,represented by a graph H in which | V H ( X ) | ≥ , and such that for everysubset Z ⊆ X , N/Z is vertically 3-connected and
H/Z is 2-connected andtype 3.Proof.
By Lemma 5.7, there are three edges a, b, c ∈ E ( G ), spanning at leastfive vertices, such that for every subset Z ⊆ { a, b, c } , G/Z is 2-connected andtype 3. Thus for each subset Z ⊆ { a, b, c } , N \ e/Z is essentially 3-connected,type 3, and of rank at least five, and so by Lemma 3.4 is represented uniquelyup to replacement by G/Z . For each subset Z ⊆ { a, b, c } , let H Z be a graphrepresenting N/Z . By Lemma 5.9,
N/Z is vertically 3-connected and H Z is2-connected and type 3. Hence by Theorem 3.2, N/Z is uniquely representedby H Z . Thus for each Z ⊆ { a, b, c } , G/Z = H Z \ e , up to replacement.For each edge z ∈ { a, b, c } the contraction G/z induces a map V ( G ) → V ( G/z ) in the obvious way: each vertex that is not an end of z maps toitself, and each vertex that is an end of z maps to the new vertex resultingfrom the identification of the ends of z in the contraction operation. Let usdenote the new vertex of G/z resulting from the contraction of the edge z by ¯ u z . Then for every vertex x ∈ V ( G ), x (cid:55)→ ¯ u z if and only if x is incidentwith z . For each z ∈ { a, b, c } , let us denote by τ z : V ( G ) → V ( G/z ) thismap defined by contracting z in G . For each z ∈ { a, b, c } , G/z = H z \ e . Letus denote by V z ( e ) = { v z , v (cid:48) z } the set of vertices of G/z that are the ends of e in H z (where possibly v z = v (cid:48) z if e is a loop). Then τ − z ( V z ( e )) is a set ofvertices of G ; let us write τ − z ( e ) for short to denote the inverse image under τ z of the ends of e in G/z . Note that for each z ∈ { a, b, c } , 1 ≤ | τ − z ( e ) | ≤ e is a loop incident to a vertex x ∈ V ( H z ) that is not the new vertex¯ u z ∈ V ( G/z ) resulting from the contraction of z , then τ − z ( e ) = { x } . If e has a pair of distinct ends x, y ∈ V ( H z ), neither of which are the new vertex¯ u z , then τ − z ( e ) = { x, y } ; if e is a loop incident to ¯ u z then τ − z ( e ) is the pairof ends of z in G . If e has ¯ u z as one end and a different vertex x ∈ V ( H z ) as its other end, then τ − z ( e ) = { v , v , x } , where v , v are the ends of z in G .Consider H a and its vertices that are the ends of e . Suppose first that¯ u a / ∈ τ − a ( e ). Then τ − a ( e ) is either a single vertex x ∈ V ( G ) or a pair ofvertices x, y ∈ V ( G ). Let H be the graph obtained from G by adding e asa loop incident to x in the first case, and as an x - y edge in the second.Suppose now that ¯ u a ∈ τ − a ( e ). Then either | τ − a ( e ) | = 2 or | τ − a ( e ) | = 3.Suppose first that | τ − a ( e ) | = 2. That is, τ − a ( e ) = { x, y } , where x, y are theends of a in G , so e is a loop incident to ¯ u a in H a . Because H ab uniquelyrepresents N/ab , and H a /b = H ab = H b /a , in H b edge e is either a loopincident an end of a or an edge sharing both ends of a . Let H be the graphobtained by adding e as an edge to G with its end or ends as in H b .Now suppose that ¯ u a ∈ τ − a ( e ) and | τ − a ( e ) | = 3. Then τ − a ( e ) = { x, y, w } ,where x, y are the ends of a in G , and while ¯ u a is one end of e in H a , theother distinct end of e in H a is w ∈ V ( G ). Consider H a /b = H ab = H b /a .If e is a loop in H ab , then in H b , e is an edge in parallel with b , so w is theother end of b in H a , and in H ab , ¯ u a = ¯ u b , so a and b are adjacent in G .Let H be the graph obtained by adding e as an edge to G in parallel with b .Assume now that e is not a loop in H ab . Then since H a /b = H ab = H b /a , in H b edge e either has x or y as one of its ends. Let H be the graph obtainedfrom G be adding e as an x - w or y - w edges according to its end incident to a in H b and its end w in H a . Claim. B ( H ) is our required twin for N .The claim follows immediately from the fact that for each subset Z ⊆{ a, b, c } , H/Z = H Z . (cid:3) Types 1 & 2.
We now show that if N is an excluded minor for theclass of bicircular matroids, and has rank at least seven, then N has anelement e such that N \ e is essentially 3-connected. We use the followingversion of Bixby’s Lemma. Theorem 5.11 (Bixby’s Lemma) . Let e be an element of a vertically 3-connected matroid M . Then either M \ e or M/e is essentially 3-connected.
Let N be an excluded minor for the class of bicircular matroids. Ourgoal is to find a triple of elements X = { a, b, c } such that for every subset Z ⊆ X , N/Z is vertically 3-connected. If N has an element e such that N \ e is essentially 3-connected and type 3, then by Lemmas 4.8 and 5.10 we aredone. So suppose N has no such element, and let e ∈ E ( N ). Then either(1) N \ e is essentially 3-connected and type 1 or type 2, or(2) N \ e has an essential 2-separation.We show that the second case does not occur. Lemma 5.12.
Let N be an excluded minor for the class of bicircular ma-troids, of rank at least eight. Let e ∈ E ( N ) and assume N/e is essentially
XCLUDED MINORS FOR BICIRCULAR MATROIDS 27 N has an element f such that N \ f is essentially 3-connected. Moreover, if N/e is type 3, then N has an element f such that N \ f is essentially 3-connected and type 3.Proof. Let G be a graph representing N/e . Since
N/e is essentially 3-connected with no non-trivial series class, G is 2-connected with minimumdegree at least 3. If G has a 2-separation, then let ( A, B ) and (
C, D ) beproper 2-separations of G with A ⊆ C and D ⊆ B with A and D minimal.By Proposition 2.6, there are edges f ∈ A and f (cid:48) ∈ D such that G \{ f, f } is2-connected. If, on the other hand, G is 3-connected, then choose an edge f ∈ E ( G ). If G \ f remains 3-connected, choose a second edge f (cid:48) : G \{ f, f (cid:48) } is2-connected. If G \ f has a 2-separation, again applying Proposition 2.6, G \ f has an edge f (cid:48) such that G \{ f, f (cid:48) } remains 2-connected. Thus N/e \{ f, f } is essentially 3-connected.If N \{ f, f (cid:48) } is essentially 3-connected, then N \ f is essentially 3-connected.Otherwise, as N \{ f, f (cid:48) } /e is essentially 3-connected, N \{ f, f (cid:48) } has a uniqueessential 2-separation ( A, B ). If both of f and f (cid:48) were spanned by one sideof the separation ( A, B ), then (
A, B ) would extend to a 2-separation of N .Since N is vertically 3-connected, this does not occur. Thus one of f or f (cid:48) , without loss of generality let us say f (cid:48) , is not spanned by A nor by B .Therefore N \ f is essentially 3-connected.Now if N/e if type 1 or 2, or if
N/e is type 3 and also N \ f is type 3,we are done. So assume that N/e is type 3 but N \ f is type 1 or type2. Suppose first N \ f has a type 1 representation H . The graph G \ f is2-connected and has at most two vertices of degree 2. Let v be the apexvertex of H . Since after possibly rolling and replacement, H/e = G \ f , G \ f is type 1 with apex v , and f does not have v as an end in G (else G wouldalready be type 1, but G is type 3). The subgraph ( G − v ) \ f has at least sixvertices and at least two leaves, and at most two vertices that have degreetwo in G \ f . If ( G − v ) \ f has just two leaves and both have degree 2 in G \ f , then each non-leaf vertex has an incident edge whose other end is v :let f (cid:48)(cid:48) be such an edge. Otherwise, G − v has a leaf x whose degree in G is at least 3, and so has at least two x - v edges: let f (cid:48)(cid:48) be one of these x - v edges. In either case, G \ f (cid:48)(cid:48) is 2-connected and type 3. We claim that N \ f (cid:48)(cid:48) is essentially 3-connected and type 3. For let K be a graph representing N \ f (cid:48)(cid:48) . Since K/e = G \ f (cid:48)(cid:48) up to replacement, K/e is type 3, and so K istype 3 (by Proposition 5.8). But suppose N \ f (cid:48)(cid:48) has an essential 2-separation( A, B ), say with e ∈ A . Since G \ f (cid:48)(cid:48) is 2-connected, N \ f (cid:48)(cid:48) /e is essentially3-connected. Thus by Proposition 2.5, K [ A ] consists of a pendant set ofedges possibly together with a set of loops incident to its pendant vertex; K [ B ] therefore contains all vertices of G . Thus f (cid:48)(cid:48) ∈ cl M ( B ). This impliesthat the essential 2-separation ( A, B ) of N \ f (cid:48)(cid:48) extends to an essential 2-separation of N . Since this essential 2-separation is a vertical 2-separation,this is a contradiction. The case that N \ f is type 2 is similar to the previous case. Let H bea type 2 graph representation of N \ f , with apex v and rotation vertex u .Up to rolling, rotation, and replacement, H/e = G \ f . Thus G \ f is type 2with apex v and rotation vertex u . The subgraph ( G − v ) \ f has at leastsix vertices, and at least one vertex of degree 1. The graph G \ f has atmost two vertices of degree 2, with all remaining vertices of degree at least3. Hence we either find a vertex different than u of ( G − v ) \ f that has atleast two incident edges whose other end is v , or a vertex different than u that is not pendant in ( G − v ) \ f but has an incident edge whose other endis v . Choose such an edge f (cid:48)(cid:48) (cid:54) = f . Then G \ f (cid:48)(cid:48) is 2-connected and type3. The rest of the argument is similar to that in the previous case. Let K be a graph representing N \ f (cid:48)(cid:48) . Since K/e = G \ f (cid:48)(cid:48) up to replacement, K/e is type 3, and so, so is K type 3. If N \ f (cid:48)(cid:48) has an essential 2-separation( A, B ), with e ∈ A , then applying Proposition 2.5 we similarly deduce that f (cid:48)(cid:48) ∈ cl M ( B ) and so that ( A, B ) extends to a vertical 2-separation of N , acontradiction. (cid:3) To complete this section, we just need to show that we can find ourdesired triple of elements a, b, c , in the case that N is an excluded minor ofsufficient rank that does not have an element e for which either N \ e or N/e is essentially 3-connected and type 3. So let N be such an excluded minor.By Lemma 5.12 and the discussion preceding, N has an element e for which N \ e is essentially 3-connected and type 1 or 2. Lemma 5.13.
Let N be an excluded minor for the class of bicircular ma-troids. Assume N has rank greater than seven, and that N does not have anelement whose deletion or contraction is essentially 3-connected and type 3.Let e be an element such that N \ e is essentially 3-connected. Then there isa bicircular twin M for N , relative to a set X of size three, represented bya graph H with | V H ( X ) | ≥ , and such that for every subset Z ⊆ X , H/Z is 2-connected, and
N/Z is vertically 3-connected.Proof.
Let G be a graph representing N \ e . Then G is type 1 or type 2.By Proposition 4.6, G has a unique apex v , and v is not an internal vertexof a line or balloon of G . By Lemma 4.5, N \ e has at most two non-trivialseries classes, and by Proposition 4.3, the non-trivial series classes of N \ e correspond precisely to the lines and balloons of G .The two cases that N \ e is type 1 or type 2 are dissimilar enough that,in the interests of presenting clean arguments, it is easier to deal with themseparately for the first part of the argument.Case 1. N has an element e such that N \ e is essentially 3-connected andtype 2. Assume G is type 2. Let u be the rotation vertex of G . Claim. If G is not 2-connected, then G has an element f such that N \ f isessentially 3-connected and represented by a 2-connected type 2 graph. XCLUDED MINORS FOR BICIRCULAR MATROIDS 29
Proof of Claim.
By replacement if necessary, we may assume each balloon of G is standard. Since G is type 2, every balloon of G has vertex of attachment v . By Lemma 4.5, if G has two balloons then each has size at most three,while if G has just one balloon it may have size at most four, and if it hassize four then G does not have a line. Thus there are at least two verticesin the unique component T of G − { u, v } that is a tree, and in any casethere is an edge f ∈ G [ V ( T ) ∪ v ], not contained in a line or balloon of G , such that co( G ) \ f is 2-connected. Note that since f is not incident to u , G \ f remains type 2. Let H be a graph representing N \ f . By Lemma4.7, G \ f = H \ e up to rotation and replacement, H is 2-connected, N \ f isessentially 3-connected, no balloon of G has size greater than three, and in H the edge e repairs the balloon or balloons of G . This G implies that G has only one balloon S .Further, this implies that S has size two and that S is a pair of paralleledges in H : for suppose to the contrary that the balloon S of G has sizethree or that H [ S ] consists of a pendant edge with a loop. If | S | = 3, H [ S ]must form a standard balloon (else in H a subset of edges of S would forma line, contrary to statement (ii) of Lemma 4.7). In either case, H is a type3 representation of the essentially 3-connected matroid N \ f , contrary toassumption. Since G \ f is type 2, so is H \ e type 2; since H [ S ] is a pair ofparallel edges both incident to v , so also H is type 2. (cid:3) By the previous claim, we may now assume that G is 2-connected. Claim.
There is a triple of edges X = { a, b, c } in G spanning at least fivevertices such that G/Z is 2-connected and type 2, for every subset Z ⊆ X . Proof of Claim.
Let w be the non-rotation vertex incident to exactly two ofthe rotation lines (or edges) of G . Since G is 2-connected and G − { u, v } isa tree, if G − u is not 2-connected, then w has degree 1 in G − u . If thisis the case, then either G − { u, w } is 2-connected or there is a line L , byLemma 4.5 of length at most three, with ends w and x ∈ V ( G − { u, v } )such that the graph G (cid:48) obtained from G by deleting u , w , and the internalvertices of L is 2-connected. Let a be the edge in G − u incident with w , andconsider the 2-connected graph G (cid:48) (where if G − { u, w } is 2-connected thenset G (cid:48) = G − { u, w } ). The graph G (cid:48) has at least four vertices and G (cid:48) − v is atree; clearly we may choose a pair of non-adjacent edges b, c of G (cid:48) such that G (cid:48) / { b, c } remains 2-connected. Let X = { a, b, c } . Then X spans at leastfive edges in G , and G/Z remains 2-connected for each Z ⊆ X .So now assume G − u is 2-connected. Write G (cid:48) = G − u . Let S denotethe set of contractable edges of G (cid:48) . Assuming that si( G (cid:48) ) is not K ,n nor K (cid:48) ,n , G (cid:48) [ S ] has a three-edge matching X = { a, b, c } , by Proposition 5.3. ByProposition 5.5 G (cid:48) /Z is 2-connected for every subset Z ⊆ X . As long asnone of a, b, c is a edge linking w and v , G/Z also remains 2-connected andis type 2, for each Z ⊆ X . But suppose one of a, b, c is a v - w edge; withoutloss of generality, suppose a = vw . Since G (cid:48) is 2-connected there is an edge a (cid:48) ∈ E ( G (cid:48) ) incident to w whose other end is not v . Let X (cid:48) = { a (cid:48) , b, c } . Thisedge a (cid:48) is contractable, and unless the three edges a (cid:48) , b, c induce a subgraphcontaining a v - w path, G/Z remains 2-connected for every Z ⊆ X (cid:48) . Sincenone of a (cid:48) , b , nor c is incident to v , this does not happen. Hence G/Z remains 2-connected and type 2 for all Z ⊆ X (cid:48) .If si( G (cid:48) ) is K ,n or K (cid:48) ,n , then choose a pair of edges a, b ∈ E ( G (cid:48) ) non-incident to v and an edge c incident to v but non-adjacent to a and non-adjacent to b . Since | V ( G (cid:48) ) | ≥
7, this is clearly possible. Set X = { a, b, c } .Clearly G/Z remains 2-connected and type 2 for each Z ⊆ X . (cid:3) Let X be a set of three edges as given by the previous claim. For eachsubset Z of X , let H Z be a graph representing N/Z . Claim.
For every subset Z of X , H Z is 2-connected and type 2 with apex v , e has v as an end, and N/Z is vertically 3-connected.
Proof of Claim.
Since
G/Z is 2-connected, N \ e/Z is essentially 3-connected.Each of G/Z and H Z \ e represent N \ e/Z ; since N \ e/Z has rank at leastfive, and G/Z is type 2, by Lemma 3.4,
G/Z = H Z \ e up to rotation andreplacement. Thus H Z \ e is type 2 with apex v and is 2-connected up toreplacement. But no replacement operation applied to a 2-connected graphyields a graph with a 1-separation, so H Z \ e is 2-connected.Suppose for a contradiction that H Z has a proper 1-separation ( A, B ),say with V ( A ) ∩ V ( B ) = { x } and e ∈ A . Then neither A nor B is a parallelclass of N/Z , and since
N/Z has no non-trivial series class, H Z has neithera balloon nor a line, so ( A, B ) is an essential 2-separation of
N/Z . Since
N/Z \ e is essentially 3-connected, by Proposition 2.5, A − e is a balloon of H Z \ e , with vertex of attachment x . But this contradicts the fact that H Z \ e is 2-connected. Because H Z is 2-connected, N/Z is essentially 3-connected.Since
N/Z has no non-trivial series class,
N/Z is vertically 3-connected.Now suppose for a contradiction that H Z is not type 2. That is, N/Z istype 1 or type 3. We have already seen that H Z \ e is type 2, so H Z cannotbe type 1. So N/Z is type 3. But
N/Z is a minor of one of
N/a , N/b , or
N/c , so by Proposition 5.8, one of these is type 3. Each of
N/a , N/b and
N/c is essentially 3-connected, so this is contrary to assumption. So H Z istype 2. Since H Z \ e is type 2 with apex v , H Z must also have apex v . Thuswere e not incident to v in H Z , H Z would be type 3. (cid:3) Case 2. N \ e is type 1. We first show that we may now further assume that N has no element e such that N \ e or N/e is essentially 3-connected and type 2.
Claim. If N has an element e for which N/e is essentially 3-connected andtype 2, then N has an element f for which N \ f is essentially 3-connectedand represented by a 2-connected type 2 graph. Proof.
Let G be a type 2 graph representing N/e , say with apex v androtation vertex u . Since N/e has no non-trivial series class, G is 2-connected XCLUDED MINORS FOR BICIRCULAR MATROIDS 31 with minimum degree at least three. The graph G − { u, v } is a tree withat least one leaf x that is not incident to u . Since deg G ( x ) is at leastthree, there are at least two v - x edges in G . Let f be a v - x edge. Then G \ f is 2-connected, and N/e \ f is essentially 3-connected. Let H be agraph representing N \ f . By Lemma 3.4, H/e = G \ f , up to rotation andreplacement.We claim N \ f is essentially 3-connected. For suppose to the contrary that N \ f has an essential 2-separation ( A, B ), say with e ∈ A . Since N \ f /e isessentially 3-connected, by Proposition 2.5, H [ A ] consists of a pendant setof edges together possibly with a set of loops incident to its pendant vertex.Contracting e in H thus yields at least one loop incident to the single vertexin V ( A ) ∩ V ( B ). Since N \ f /e is type 2, represented by G \ f , this implies V ( A ) ∩ V ( G ) = { v } . But it is clear that f is in cl N ( B ), so the 2-separation( A, B ) of N \ f extends to a vertical 2-separation ( A, B ∪ f ) of N , contraryto Lemma 2.2. Thus H is 2-connected and N \ f is essentially 3-connected.By assumption, N has no element whose deletion or contraction is essen-tially 3-connected and type 3. Thus N \ f , and H , is either type 1 or type 2.But H/e = G \ f up to rotation and replacement, and G \ f is type 2, so H/e is type 2. If H is type 1, then H/e cannot be type 2. Thus H is type 2. (cid:3) By the previous section and the previous claim, we may now assume that N does not have an element e for which N \ e or N/e is essentially 3-connectedand type 2 or type 3.Let us assume that G is substandard; that is, every balloon of G has v asits vertex of attachment and every loop not properly contained in a balloonis incident to v . Claim. If G is not 2-connected, then G has an element f such that N \ f is essentially 3-connected and represented by a 2-connected graph, up torolling. Proof of Claim.
Suppose G is not 2-connected. Since G is substandard, thisimplies that G has a 1-separation ( A, B ) with V ( A ) ∩ V ( B ) = { v } , in which G [ A ] consists of one or two balloons and G [ B ] − v is a tree T . The component T of G − v has at least two leaves, at most one of which may be an internalvertex of a line, by Lemma 4.5. Let u be a leaf of T that is not an internalvertex of a line. Then there are at least two v - u edges in G . Let f be a v - u edge. Then f is not contained in a line or balloon of G and co( G ) \ f remains 2-connected. Let H be a graph representing N \ f . By Lemma 4.7, N \ f is essentially 3-connected, and H is 2-connected up to rolling. (cid:3) By the claim, we may now assume that G is 2-connected. Claim.
There is a triple of edges X = { a, b, c } in G spanning at least fivevertices such that G/Z is 2-connected, for every subset Z ⊆ X . Proof of Claim.
Let S denote the set of contractable edges of G . Assumingthat si( G ) is not K ,n nor K (cid:48) ,n , G has a three-edge matching X = { a, b, c } , by Proposition 5.3. By Proposition 5.5 G/Z is 2-connected for every subset Z ⊆ X .So now assume that si( G ) is K ,n or K (cid:48) ,n . Then the apex vertex v of G isone of the two vertices of si( G ) of degree n or n + 1; let u be the other vertexof degree n or n + 1 in si( G ), and let Y be the set of remaining vertices of G .Let a = uy and b = uy be a pair of vertices sharing u as one end and witha distinct ends y (cid:54) = y in Y , and let c be a edge with ends v and y ∈ Y ,with y / ∈ { y , y } . As | Y | ≥
6, this is clearly possible. It is also clear that
G/Z remains 2-connected and type 1 for every subset Z ⊆ X . (cid:3) For each subset Z of X , let H Z be a graph representing N/Z . Claim.
For every subset Z of X , H Z is 2-connected and type 1 with apex v , up to rolling e has v as an end, and N/Z is vertically 3-connected.
Proof of Claim.
Since
G/Z is 2-connected, N \ e/Z is essentially 3-connected.Each of G/Z and H Z \ e represent N \ e/Z ; since N \ e/Z has rank at leastfive, and G/Z is type 1, by Lemma 3.4,
G/Z = H Z \ e up to rolling andreplacement. Thus H Z \ e is type 1 with apex v and 2-connected up torolling.Now suppose for a contradiction that H Z has a proper 1-separation ( A, B ),say with V ( A ) ∩ V ( B ) = { x } and e ∈ A . Then neither A nor B is a parallelclass of N/Z . Since
N/Z has no non-trivial series class, H Z has no balloonand no line, and ( A, B ) is an essential 2-separation of
N/Z . Since
N/Z \ e is essentially 3-connected, by Proposition 2.5, A − e is a balloon in H Z \ e ,with vertex of attachment x . This implies e ∈ cl N/Z ( A − e ). Since G/Z is2-connected up to rolling, x (cid:54) = v (that is, the balloon A − e in H Z \ e does nothave v as its vertex of attachment). Thus A − e is a line of G/Z with verticesof attachment x and v , and so A − e is a line of G . Thus A − e is a seriesclass of N/Z if and only if it is a series class of N , and so e ∈ cl N/f ( A − e )if and only if e ∈ cl N ( A − e ). Thus the 2-separation ( A, B ) of
N/Z extendsto a 2-separation (
A, B ∪ Z ) of N , contrary to Lemma 2.2.Because H Z is 2-connected, N/Z is essentially 3-connected. Since
N/Z has no non-trivial series class,
N/Z is vertically 3-connected.Now suppose for a contradiction that H Z is not type 1. That is, N/Z istype 2 or type 3. But this is impossible, since by assumption N does nothave an element f for which N/f is essentially 3-connected and type 2 ortype 3. Since
N/z is type 1 for each z ∈ { a, b, c } , N/Z is also type 1. Since H Z \ e is type 1 with unique apex v , H Z must also have apex v . If in H Z edge e had two distinct ends neither of which were v , then H Z would betype 3, a contradiction. (cid:3) Constructing a bicircular twin for N . We now have, in each of cases 1 and 2 above, a 2-connected graph G representing N \ e , a set X = { a, b, c } ⊆ E ( G ) such that for each subset Z of X , N/Z is vertically 3-connected. In the case that G is type 2, N/Z istype 2 for each Z ⊆ X , and when G is type 1, so is N/Z . XCLUDED MINORS FOR BICIRCULAR MATROIDS 33
We now construct a bicircular twin for N relative to X = { a, b, c } . Justas in the proof of Lemma 5.10, we use the “contraction map”: for each z ∈{ a, b, c } the contraction G/z induces a map V ( G ) → V ( G/z ) in the obviousway: each vertex that is not an end of z maps to itself, and each vertex thatis an end of z maps to a new vertex resulting from the identification of theends of z as a result of the contraction operation. For each z ∈ { a, b, c } ,denote this map by τ z : V ( G ) → V ( G/z ). Denote the new vertex formedby contracting the edge z by ¯ u z , so for any vertex x ∈ V ( G ), x (cid:55)→ ¯ u z if andonly if x is an end of z . For each z ∈ { a, b, c } , G/z = H z \ e , and (assumingsubstandard representations in the case that H z is type 1) in H z , the edge e has v as one end. Denote by v a , v b , and v c , respectively, the vertex of G/a , G/b , G/c , resp., that is the second end of e in H a , H b , and H c , resp., whereif e is a loop incident to v in one of these graphs then we consider v to bethe second end of e (so for instance if e is a loop incident to v in H a , then v a = v ). As long as v z (cid:54) = ¯ u z then v z is a vertex of G .If v a = v b = v c , then in particular there is no z ∈ { a, b, c } for which v z = ¯ u z , and so there is a vertex y ∈ V ( G ) not incident to any of a, b, c , suchthat y = v a = v b = v c . Let H be the graph obtained by adding e as an edgeto G linking v and y (where if y = v , then e is a loop incident to v ).Assume now that it is not the case that v a = v b = v c ; without loss ofgenerality assume that v a (cid:54) = v b . Claim. If v a (cid:54) = v b , then up to relabelling a and b , either(1) in G , a and b are non-adjacent, v a is an end of b , and and v b = ¯ u b .(2) a and b are adjacent in G , v a = ¯ u a and v b = ¯ u b (3) in G , a and b are adjacent, v a = ¯ u a , v b (cid:54) = ¯ u b , and v b is an end of a . Proof of Claim.
This follows from the observation that H a /b = H ab = H b /a ,and so it must be the case that v a and v b are identified in H ab to a singlevertex resulting from the contraction operations in each graph. (cid:3) We define a graph H according to the possible outcomes when v a (cid:54) = v b . • Suppose the first outcome of the claim holds. Since H a /c = H ac = H c /a , v c = v a . Let H be the graph obtained by adding e as an edgewith ends v and v a . • Suppose the second outcome of the claim holds. Since a and b areadjacent, c is not adjacent to a nor b . The fact that H a /c = H c /a implies that v c is one of the ends of a in G . The fact that H b /c = H c /b implies that v c is one of the ends of b in G . Thus v c must bethe common end x of a and b in G . Let H be the graph obtained byadding e as an edge with ends v and x . • Suppose the third outcome of the claim holds. As in the previouscase, c is adjacent to neither a nor b . Since H b /c = H bc = H c /b , v c = v b . Let H be the graph obtained by adding e as an edge withends v and v b . Claim. B ( H ) is a twin for N relative to X . Proof of Claim.
For each z ∈ { a, b, c } , H/z = H z . Thus for each Z ⊆{ a, b, c } , H/Z = H Z , and so B ( H/Z ) =
N/Z . (cid:3) This completes the proof of Lemma 5.13. (cid:3) No excluded minor has rank greater than seven
Quasi-graphic matroids.
For working with quasi-graphic matroidsand finding minors, it is convenient to have additional information aboutthe framework of a quasi-graphic matroid. For this we use the results of [1].Let M be a quasi-graphic matroid, and let G be a framework for M . Setting B = { C : C is a cycle of G and a circuit of M } we obtain a biased graph( G, B ) from the framework G . The collection B tells us which cycles of G are circuits of M , but this does not determine M . A bracelet in a biasedgraph is a pair of vertex-disjoint unbalanced cycles. In general, a braceletin ( G, B ) may or may not be a circuit of M .A bracelet function is a function from the set of bracelets of a biased graph( G, B ) to the set { independant , dependant } . Given a biased graph ( G, B ), the bracelet graph B ( G, B ) of ( G, B ) is the graph with vertex set the collectionof bracelets of ( G, B ) in which two bracelets are joined by an edge if andonly if their union has the property that the minimum number of edgesthat must be removed to obtain an acyclic subgraph is exactly three. If χ is a bracelet function with the property that χ ( Y ) = χ ( Y (cid:48) ) whenever Y and Y (cid:48) are bracelets in the same component of B ( G, B ), then χ is a proper bracelet function. Given a biased graph ( G, B ) with bracelet function χ , let C ( G, B , χ ) be the collection of edge sets of: balanced cycles, thetas with nocycle in B , tight handcuffs, bracelets Y with χ ( Y ) = dependant , and loosehandcuffs containing bracelets Y with χ ( Y ) = independant . The followingtwo theorems are proved in [1]. Theorem 6.1 ([1], Theorem 2.1) . Let ( G, B ) be a biased graph with G connected, and let χ be a bracelet function for ( G, B ) . If C ( G, B , χ ) is theset of circuits of a matroid, then χ is proper. Theorem 6.2 ([1], Theorem 1.1) . Let M be a matroid and let ( G, B ) be abiased graph with E ( G ) = E ( M ) . The following are equivalent. (i) There is a proper bracelet function χ for G such that M = M ( G, B , χ ) . (ii) M is quasi-graphic with framework G and B is the set of cycles of G that are circuits of M . We say the quasi-graphic matroid M = M ( G, B , χ ) is represented by thetriple ( G, B , χ ) consisting of the biased graph ( G, B ) with bracelet function χ . It is shown in [3] that if M is quasi-graphic with framework G , then M \ e and M/e are quasi-graphic with framework G \ e and G/e , respectively,where if e is an unbalanced loop then G/e is the graph G ◦ e defined in Section3.2 for contracting an unbalanced loop in a biased graph. It is shown in [1]that if M is represented by ( G, B , χ ), then M ( G, B , χ ) \ e and M ( G, B , χ ) /e XCLUDED MINORS FOR BICIRCULAR MATROIDS 35
12 34 56 1 2 4 5612 3 4 56 3 1 2 4 56 3
Figure 6. M (2 C ) and M ( K )
123 4 56 7 23 71 45 6 2 3 7 1456
Figure 7. Pf are represented by naturally defined biased graphs ( G, B ) \ e , ( G, B ) /e alongwith their inherited bracelet functions. For details, see Sections 2 and 4.2of [1].6.2. Six small excluded minors.
In the following section, we show thatthere is no excluded minor of rank greater than seven by showing that anysuch purported matroid in fact already contains one of six smaller excludedminors. In this section we describe these matroids. Each is quasi-graphic,and so has a biased graphic representation. Some have more than one suchrepresentation; here we exhibit those representations that are useful for find-ing these matroids as minors in the proof of Theorem 6.5. Geometric repre-sentations and frameworks are shown in Figures 6-10. Points in geometricrepresentations are shown as solid discs, while vertices of graphs are opencircles.The graph 2 C is obtained by adding an edge in parallel to each of theedges of a 3-cycle. Its cycle matroid M (2 C ) has rank two and consistsof three 2-element rank-1 flats, one of which is a balanced 2-cycle in theframework shown in Figure 6. At right in Figure 6 is a framework for M ( K ), with balanced cycle { , , } . The matroid Pf is shown in Figure 7,along with two of its frameworks. The framework shown at centre in Figure7 has a single balanced cycle { , , , } , and the framework at right hasdependant bracelet { , } ∪ { , } . The framework for P shown in Figure8 has a single balanced cycle { , , , } . The framework for P shown inFigure 9 has the single balanced cycle { , , , } . The matroid M { , , , , } .
123 4 56 7 23 7 1456
Figure 8. P
12 1 27 734 56 345 6
Figure 9. P
123 4 5678dual 12 3 4 5 678
Figure 10. M No excluded minor has rank greater than seven.
Let N and M be matroids on a common ground set E . We say that a set Y ⊆ E is a disagreement set if r N ( Y ) (cid:54) = r M ( Y ). Proposition 6.3. If Y is a minimal disagreement set for N and M , then Y is a circuit in one of N or M and an independant set in the other.Proof. Since these ranks of N and M disagree on Y , we may choose C ⊆ Y so that C is a circuit of exactly one of N or M ; let us say C is a circuit of N .If C is independant in M , then C is a disagreement set. Since every subsetof C is independant in N and M , C = Y and we are done. Otherwise, C XCLUDED MINORS FOR BICIRCULAR MATROIDS 37 is dependant in M , and so there is a subset C (cid:48) of C so that C (cid:48) is a circuitof M . Since C is not a circuit in both N and M , C (cid:48) is a proper subset of C . Thus C (cid:48) is independant in N , and so C (cid:48) is a disagreement set properlycontained in Y , a contradiction. (cid:3) Proposition 6.4.
Let N and M be matroids on a common ground set E .Assume N and M are twins relative to X ⊆ E , and that Y ⊆ E is a circuitin N but independant in M . (i) No element of X is in the span of Y in N . (ii) Every element of X is in the span of Y in M . (iii) Y is a circuit in M/e whenever e ∈ X .Proof. Let e ∈ X . Suppose, contrary to (i), that e ∈ cl N ( Y ). Then r M/e ( Y ) = r N/e ( Y ) = r N ( Y ) −
1. But r N ( Y ) = r M ( Y ) −
1, so this im-plies that r M ( Y ) − r M/e ( Y ), a contradiction.Now suppose, contrary to (ii), that e / ∈ cl M ( Y ). Then r N/e ( Y ) = r M/e ( Y ) = r M ( Y ). But r M ( Y ) = r N ( Y ) + 1, so this implies that r N/e ( Y ) = r N ( Y ) + 1,a contradiction.Statement (iii) follows from statement (i): since e is not in cl N ( Y ), Y remains a circuit in N/e = M/e . (cid:3) Note that statement (i) of Proposition 6.4 implies that no minimal dis-agreement set for N and M contains an element of X . Theorem 6.5.
Let N and M be matroids on a common ground set E . Let X = { a, b, c } ⊆ E be a subset of three elements such that (i) N and M are twins relative to X , (ii) For every subset Z of X , N/Z = M/Z is vertically 3-connected, and (iii) M is bicircular, represented by a graph G in which | V G ( X ) | ≥ .Then N contain one of M (2 C ) , M ( K ) , P , Pf , P , or M as a minor.Proof. We proceed via a series of three claims. The first two will establishthat N is quasi-graphic. The third claim is that N either contains Pf or isframe. Then from the graph representation of M we obtain a biased graphrepresenting N as a frame matroid. Finally, we show that this biased graphcontains a biased graph representation of one of the matroids listed as aminor. Claim . A minimal disagreement set is a circuit in N and independantin M . Proof of Claim.
Suppose to the contrary that a minimal disagreement set Y is a circuit in M and independant in N . Then G [ Y ] forms either handcuffsor a theta. By Lemma 6.4, no element of X is in cl M ( Y ), so no edges in X has both ends in V G ( Y ). By Lemma 6.4, each element of X is in cl N ( Y ).In particular, b and c are in the span of Y in M/a = N/a . Thus in
G/a ,both ends of b and c are in V G/a ( Y ). This is is only possible if a , b , and c have a common end in G . But this is contrary to the assumption that | V G ( X ) | ≥ (cid:3) Claim . A minimal disagreement set is either a cycle or a pair of vertexdisjoint cycles in G . In the first case each of a, b, c is a chord of the cycle,while in the second case each of a, b, c has one end in each cycle. Proof of Claim.
Let Y be a minimal disagreement set. By Lemma 6.4, eachof a, b, c are in the span of Y in M , and while Y is independant in M , Y isa circuit in M/z for each z ∈ X . Thus Y ∪ { z } is a circuit of M , for each z ∈ X . Suppose G [ Y ] contains a vertex u of degree 1. Because | V G ( X ) | ≥ z ∈ X that is not incident with u . But then the subgraphof G/z induced by Y still has u as a vertex of degree 1, and so is not acircuit of M/z , a contradiction. Thus G [ Y ] has minimum degree 2. But Y is independant in M , so G [ Y ] forms a collection of disjoint cycles. But foreach z ∈ X , Y is a circuit of M/z , and so induces a bicycle in
G/z . Thuseither G [ Y ] is a cycle and each of a , b , and c is a chord of G [ Y ], or G [ Y ] is apair of disjoint cycles and each of a , b , and c have one end in each cycle. (cid:3) Put B = { C : C is a circuit of N and a cycle of G } . Claims 6.5.1 and6.5.2 imply that ( G, B ) is a biased graph. Let χ be the bracelet functionmapping a bracelet Y of ( G, B ) to dependant if and only if Y is a circuit of N . Claims 6.5.1 and 6.5.2 along with Theorems 6.1 and 6.2 imply that N is quasi-graphic, represented by ( G, B , χ ). Claim . Either N contains Pf as a minor, or N is a frame matroidrepresented by the biased graph ( G, B ), where B = { C : C is a cycle of G and a circuit of N } . In the latter case, no cycle in B contains an edge in X ,but each edge in X is a chord of every cycle in B . Proof of Claim.
If every minimal disagreement set for N and M induces acycle in G , then χ maps every bracelet to independant , so M ( G, B , χ ) = F ( G, B ); that is, N is a frame matroid represented by the biased graph( G, B ). Otherwise, there is a bracelet Y mapped to dependant by χ . Let C , C be the pair of disjoint cycles whose union is G [ Y ]. Each edge z ∈ X has one end in V ( C ) and one end in V ( C ), and | V G ( X ) | ≥
5. Since nominimal disagreement set contains a , b , or c , the subgraph H induced by X ∪ Y contains no cycle that is a circuit of N . Thus the biased subgraph of ( G, B )induced by X ∪ Y contains no balanced cycle. Writing H = G [ X ∪ Y ], wefind as a minor of N the matroid with quasi-graphic representation ( H, ∅ , χ (cid:48) ),where χ (cid:48) is the bracelet function assigning dependant to the bracelet Y in( H, ∅ ) (Figure 11). Now it is easy to see that ( H, ∅ , χ (cid:48) ) contains a quasi-graphic representation of Pf as a minor. So N contains Pf as a minor. (cid:3) We may now assume that N is a frame matroid, represented by the biasedgraph ( G, B ) as described in Claim 6.5.3. Choose a cycle C in B . By Claim6.5.3, none of a , b , nor c are contained in C but each of a , b , and c is a chordof C .If two of the elements in X , say a and b , are non-adjacent (non-loop)edges whose ends alternate in the cyclic order on V ( C ), then ( G, B ) contains XCLUDED MINORS FOR BICIRCULAR MATROIDS 39
Y abc
Figure 11.
Finding Pf as a minor. −→ Figure 12 a biased graph representing either P or Pf as a minor (see Figures 7 and8). Otherwise, if a, b, c form a matching in G then ( G, B ) contains a biasedgraph representing either P or K as a minor. So assume no two of a, b, c have their ends alternating in the cycle order on V ( C ) and two of a, b, c areadjacent. We find as a minor of ( G, B ) either a biased graph representing M N/z is vertically 3-connected for each z ∈ X , the graph G/z is 2-connectedfor each z ∈ X . It follows that G contains a path P linking a pair of verticesof C as shown in one of the graphs at centre or right in Figure 12. By Claim6.5.3 every cycle of C ∪ X ∪ P that contains an edge in X is unbalanced,and every cycle in C ∪ P that does not have an edge in X as a chord isunbalanced. Thus the biased subgraph of ( G, B ) induced by C ∪ X ∪ P contains a biased graph representing P as a minor. (cid:3) We can now prove the main result of this section.
Theorem 6.6.
Let N be an excluded minor for the class of bicircular ma-troids. Then N has rank less than eight.Proof. Suppose to the contrary that N has rank greater than seven. Let e ∈ E ( N ). By Lemma 2.2, N is vertically 3-connected, so either N \ e or N/e is essentially 3-connected, by Bixby’s Lemma. Thus by Lemma 5.12, if N \ e is not essentially 3-connected, then N has an element e (cid:48) for which N \ e (cid:48) is essentially 3-connected. Moreover, Lemma 5.12 guarantees that if N \ e istype 3, then there is an element e (cid:48) for which N \ e (cid:48) is essentially 3-connectedand type 3.So suppose first that N has an element e for which N \ e is essentially3-connected and type 3. By Lemma 4.8 we may further assume that N \ e is represented by a 2-connected type 3 graph. Thus by Lemma 5.10, thereis a bicircular twin M for N relative to a set X of size three, satisfying theconditions of Theorem 6.5. Thus N contain one of M (2 C ), M ( K ), P , Pf , P , or M e ∈ E ( N ) for which N \ e isessentially 3-connected, N \ e is type 1 or type 2. Moreover, by Lemma 5.12, N does not have an element whose contraction is essentially 3-connected andtype 3. By Bixby’s Lemma and Lemma 5.12, N has an element e for which N \ e is essentially 3-connected. Thus by Lemma 5.13, there is a bicirculartwin M for N relative to a set X of size three satisfying the conditions ofTheorem 6.5. So again, we find N contains one of M (2 C ), M ( K ), P , Pf , P , or M (cid:3) There are only a finite number of low-rank excluded minors
In this section we prove the following.
Theorem 7.1.
There are only a finite number of excluded minors for theclass of bicircular matroids that have rank less than eight.
A pair of elements e, f of a matroid M are clones if the bijection from E ( M ) to E ( M ) that exchanges e and f while leaving every other elementfixed is an automorphism of M . We also say that the set { e, f } is a clonalpair . The ‘clone’ relation is clearly an equivalence relation on the groundset of a matroid; a clonal class of a matroid is an equivalence class underthis relation. The following useful characterisation of clonal pairs followsimmediately from the definition. Proposition 7.2.
A pair of elements { e, e (cid:48) } are clones if and only if forevery circuit C with | C ∩ { e, e (cid:48) }| = 1 , the symmetric difference C (cid:52){ e, e (cid:48) } isa circuit. A flat of a matroid is cyclic if it is a union of circuits. By Proposition 7.2, e and e (cid:48) are clones if and only if every cyclic flat of M that contains one of e or e (cid:48) contains both e and e (cid:48) .A line in a matroid is a rank-2 flat. A line is non-trivial if it containsat least three rank-1 flats. Let M be a bicircular matroid represented bythe graph G , and let L be a non-trivial line of M . Then the subgraph G [ L ]induced by L consists of a pair of vertices u, v together with all edges bothof whose ends are in { u, v } . We say a line is adjacent to another line if theirunion has rank 3. If L and L (cid:48) are a pair of adjacent lines of M , then theset of elements in L ∩ L (cid:48) is precisely the set of loops incident to the vertex x , where V G ( L ) ∩ V G ( L (cid:48) ) = { x } . Given a non-trivial line L in a bicircularmatroid represented by the graph G , let us call the unique pair of vertices u, v in the subgraph of G induced by L the ends of the line L in G .Call an element of a non-trivial line lonely if it is not also contained in anadjacent line nor properly contained in a rank-1 flat. Observe that if M is abicircular matroid, represented by the graph G , then a set of at least three XCLUDED MINORS FOR BICIRCULAR MATROIDS 41 edges sharing a common pair of distinct ends are contained in a non-trivialline and each edge in this set is lonely.
Proposition 7.3.
Let L be a non-trivial line in a matroid M and let e and e (cid:48) be two lonely elements in L . Then e and e (cid:48) are clones.Proof. Every cyclic flat of M that contains one of e or e (cid:48) contains both e and e (cid:48) . (cid:3) Proposition 7.4.
Let L be a non-trivial line in a matroid M and let e, e (cid:48) ∈ L be lonely elements of L . If M \ e is bicircular, represented by the graph G , then M \ e (cid:48) is bicircular and is represented by the graph G (cid:48) obtained byrelabelling e (cid:48) in G by e .Proof. By Proposition 7.3, e and e (cid:48) are clones. (cid:3) Lemma 7.5.
Let N be an excluded minor of rank at least three. Then N does not have as a restriction a line containing more than three lonelyelements.Proof. Suppose to the contrary that N has a line L containing at least fourlonely elements a, b, c, d . Let G a be a graph representing N \ a . Since L − a is a non-trivial line of N \ a , G a [ L − a ] consists of a pair of vertices u, v together with all edges both of whose ends are in { u, v } . Since none of b, c, d are contained in an adjacent line of N \ a , none of b, c, d are loops unless E ( L − a ) is a pendant set of edges, in which case at most one of these edgesmay be a loop incident to the pendant vertex; if this is the case then applya rolling operation so that all edges in E ( L − a ) are u - v edges. Let G b bethe graph obtained from G a by relabelling edge b by a , and let G c be thegraph obtained from G a by relabelling edge c by a . By Proposition 7.4, G b represents N \ b and G c represents N \ c .Let G be the graph obtained from G a by adding a as an edge linking theends of L , in parallel with edges b , c , and d . Consider the circuits of N and B ( G ): • { a, b, c } is a circuit of both N and B ( G ); • each circuit of N that contains at most two of a , b , or c , is a circuitof one of B ( G a ), B ( G b ), or B ( G c ); since each of G a , G b , and G c isequal to the restriction of G to their respective ground sets, eachsuch circuit is a circuit-subgraph of G ; and, • conversely, every circuit-subgraph of G containing at most two of a , b , or c is a circuit-subgraph of one of G a , G b , or G c , and so is acircuit of N .Thus the circuits of N and B ( G ) coincide, so G represents N , a contradic-tion. (cid:3) Lemma 7.6.
Let N be an excluded minor with rank at least three. Then N contains at most three pairs of elements in parallel. Proof.
Suppose to the contrary that N contain four pairs of elements inparallel, say { e, e (cid:48) } , { f, f (cid:48) } , { g, g (cid:48) } , and { h, h (cid:48) } . Let G e be a graph repre-senting N \ e . Because { f, f (cid:48) } , { g, g (cid:48) } , and { h, h (cid:48) } remain parallel pairs in N \ e , each pair is represented by a pair of loops incident to a vertex of G e ;because the three pairs are distinct parallel classes, the pairs are incidentto distinct vertices. Thus G e is a type 3 representation for N \ e . The edge e (cid:48) is not a loop in G e , because if so, say incident to vertex v , then adding e as a loop incident to v would yield a bicircular graph representation for N . Since N \ e is vertically 3-connected, G e has no lines or balloons, andsince G e has loops, G e is not in G . Thus by Theorem 3.3, G e is the uniquebicircular representation for N \ e .Let G f be a graph representing N \ f . Applying the argument of theprevious paragraph, G f is the unique bicircular representation for N \ f , andeach pair { e, e (cid:48) } , { g, g (cid:48) } , and { h, h (cid:48) } is represented as a pair of loops, no pairincident to the same vertex as another, while f (cid:48) is not a loop.Now consider N \{ e, f } . Because e and f are each contained in a non-trivial parallel class of N , N \{ e, f } remains vertically 3-connected. More-over, each of G e \ f and G f \ e contain loops incident to at least three differentvertices (namely, g and g (cid:48) , h and h (cid:48) , along with either f (cid:48) or e (cid:48) ), so each isa type 3 representation for N \{ e, f } . Since N \{ e, f } remains vertically 3-connected, neither graph contain a line or a balloon. Thus, since neither G e \ f nor G f \ e are in G , by Theorem 3.3, G e \ f = G f \ e . But this is impos-sible, because e (cid:48) is not a loop in G e \ f while e (cid:48) is a loop in G f \ e . (cid:3) Lemma 7.7.
Let N be a rank- r excluded minor for the class of bicircularmatroids. Then | E ( N ) | ≤ (cid:0) r (cid:1) + r + 4 .Proof. Arbitrarily choose an element e ∈ E ( N ), and let G be a graph rep-resenting N \ e . Then | V ( G ) | = r . By Lemma 7.5, N does not have a linecontaining more than three lonely elements, so neither does N \ e . Thus G has no more than three edges linking each pair of its vertices. By Lemma 2.2every parallel class of N has at most two elements, no vertex of G has morethan two incident loops. By Lemma 7.6, N has no more than three pairsof elements in parallel, so N \ e has no more than three pairs of elements inparallel. Thus G has at most three vertices that have two incident loops.Thus | E ( G ) | ≤ (cid:0) r (cid:1) + r + 3, and so | E ( N ) | ≤ (cid:0) r (cid:1) + r + 4. (cid:3) Theorem 7.1 now follows easily.
Proof of Theorem 7.1.
All non-empty rank-0 and all rank-1 matroids arebicircular. A rank-2 matroid is bicircular if and only if it has at most tworank-1 flats of size greater than one. It is easy to see that a rank-2 matroidhas more than two rank-1 flats containing at least two elements if and onlyif it contains M (2 C ) as a minor. So M (2 C ) is the only excluded minor ofrank 2. For r ∈ { , , . . . , } , apply Lemma 7.7: an excluded minor of rank r for the class of bicircular matroids has at most 3 (cid:0) r (cid:1) + r + 4 elements. (cid:3) XCLUDED MINORS FOR BICIRCULAR MATROIDS 43
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