Minimum Spanning Tree Cycle Intersection Problem
aa r X i v : . [ c s . D M ] F e b Minimum Spanning Tree Cycle Intersection Problem
Manuel Dubinsky
Ingenier´ıa Inform´aticaDpto. Tecnolog´ıa y Administraci´onUniversidad Nacional de AvellanedaArgentina
C´esar Massri
Dpto. de Matem´aticaUniversidad de CAECE,IMAS, CONICETArgentina
Gabriel Taubin
School of Engineering,Brown University,USA
Abstract —Consider a connected graph G and let T be aspanning tree of G . Every edge e ∈ G − T induces a cycle in T ∪ { e } . The intersection of two distinct such cycles is theset of edges of T that belong to both cycles. We considerthe problem of finding a spanning tree that has the leastnumber of such non-empty intersections.
1. Introduction
In this article we present what we believe is a novelproblem in graph theory, namely the Minimum SpanningTree Cycle Intersection (
MSTCI ) problem.The problem can be expressed as follows. Let G bea graph and T a spanning tree of G . Every edge e ∈ G − T induces a cycle in T ∪ { e } . The intersection of twodistinct such cycles is the set of edges of T that belong toboth cycles. Consider the problem of finding a spanningtree that has the least number of such pairwise non-emptyintersections.The problem arose while investigating a (yet unpub-lished) method for mesh deformation in the area of digitalgeometry processing , see [Botsch, 2010]. The method re-quires to solve a linear system and the sparsity of thematrix is related to the solution of this problem.The remaining of this section is dedicated to expressthe MSTCI problem in the context of well establishedtheories to motivate other points of view. Section 2 setssome notation and convenient definitions. In Section 3the complete graph case is analyzed. Section 4 presentsa variety of interesting properties, and a conjecture in theslightly general case of a graph (not necessarily complete)that admits a star spanning tree. Section 5 explores progra-matically the space of spanning trees to provide evidencethat the conjecture is well posed. Section 6 collects theconclusions of the article. Intersection graph theory is a longstanding and cen-tral area of graph theory covered in important textbooks[McKee and McMorris, 1999]. It is concerned with thestudy of intersection graphs .Let F = { S , . . . , S k } be any family of sets. The intersection graph denoted Ω( F ) is the graph having F as vertex set with S i adjacent S j if and only if i = j and S i ∩ S j = ∅ .Let G = ( V, E ) be a graph and F = { S , . . . , S k } be the family of edge sets corresponding to the cycles of G . Then Ω( F ) has as vertex set the cycles of G and twovertices are adjacent if the corresponding cycles share atleast one edge. In this setting, let T be a spanning treeof G and G − T ⊂ E the set of complementary edges of T . An edge e ∈ G − T induces a cycle in T ∪ { e } whichobviously is a cycle of G . So T has a canonical mappingto some subgraph of Ω( F ) . The MSTCI problem can beexpressed in the following terms: find the spanning tree T such that it maps to the sparsest possible subgraph in Ω( F ) . The classical matroid theory as developed by Tuttein [Tutte, 1965] is a fundamental theoretical toolbox withvery deep insights in graph theory.The familiy of sets described in the previous subsec-tion closely resembles the polygon matroid . An interestingformulation of the MSTCI problem can be expressed interms of its dual matroid B ( G ) , namely the bond matroid .The bond matroid can be defined as follows. Let G =( V, E ) be a connected graph. The atoms of B ( G ) are theedge subsets A ⊂ E such that G − A determine twoconnected components G and G and such that everyedge of A has one end in each component.A dendroid D of a matroid M is a set that intersectsall the atoms and is minimal, meaning that if we deletean element of D then there exists an atom A ∈ M suchthat A ∩ D = ∅ . The dendroids of B ( G ) are the sets ofedges of the spanning trees of G .Let T be a spanning tree of G . So the edges of T define a dendroid D of B ( G ) . Note that for every edge e ∈ T , T − { e } determine two subtrees that span twoconnected components G and G of G . So there is anatural injective map between the edges of T and theatoms of B ( G ) : φ T : E ( T ) → B ( G ) A where A is the atom corresponding to the set of edgeslinking G and G . As a remark, note that e ∈ A . In thelanguage of matroid theory this set of | V | − atoms (ie.the image of φ T ) of B ( G ) is called the dendroid basis determined by D .To formulate the MSTCI problem in this frameworkwe have to be precise about the pairwise intersectionof cycles. In this sense let T be a spanning tree of aconnected graph G = ( V, E ) . Let S be the dendroid basisdetermined by the edges of T and A ∈ S an atom. Clearlythere exist a unique edge e ∈ T such that φ T ( e ) = A .Note that each edge e ′ ∈ A − { e } determine a cycle c ∈ T ∪ { e ′ } and that e ∈ c . So two such cycles havenon-empty intersection. It is not difficult to realize thatevery non-empty pairwise intersection is of this form. Ifwe manage to count the set of this pair of edges of all theatoms of S we could express the MSTCI problem as analternative minimization problem: find the spanning treesuch that its corresponding dendroid basis has the leastnumber of such pair of edges. The importance of homological methods [Cartan, 1956] in topology and geometry cannot beoveremphasized. These methods constitute fundamentalalgebraic tools that enable the computation of invariantquantities of spaces. In its original form the maininvariant was the number of “holes” of a space known asits
Betti numbers .We introduce some elementary notions based thesenotes [Dewan, 2010]. Let R be a commutative ring, andsuppose we have a sequence of R -modules M i and ho-momorphisms d i . . . d −→ M d −→ M d −→ M d −→ M − d − −−→ . . . such that ∀ i d i ◦ d i +1 = 0 . Such a sequence is called a chain complex of R -modules, and denoted M • . Becausethe composition of adjacent homomorphisms is trivial, im ( d i +1 ) ⊆ ker ( d i ) . Therefore, for each module in thechain there is a quotient H i ( M • ) = ker ( d i ) im ( d i +1 ) called the i-th homology group of M • .In graph theory this definitions become very con-crete. Let R be a principal ideal domain (ie. Z ) and let G = ( V, E ) be a graph. Consider an arbitrary orientationof the edges that maps every edge e ∈ E to a triple: e ( e, s, t ) , where s, t ∈ V are the source and target endsof the orientation of e . Let M be the free R -module over V (formal linear combinations of the vertices of G ). And M the R -module generated by the oriented edges subjectto the relations ( e, s, t ) + ( e, t, s ) = 0 (where ( e, t, s ) expresses de traversal of the edge ( e, s, t ) in the oppositedirection). Now consider the boundary homomorphism ∂ : M → M defined on the generators as ∂ ( e, s, t ) = t − s . Notsurprisingly ker ( ∂ ) is denoted the cycle space of G , C ( G ) . Since if we consider in M the linear combinationrepresenting a cycle c of G then ∂ ( c ) = 0 . It is not difficultto check that rank ( C ( G )) = | E | − | V | + c where c isthe number of connected components of G . Now we candefine the following chain complex G • : . . . d =0 −−−→ d =0 −−−→ M d = ∂ −−−→ M d =0 −−−→ d − =0 −−−−→ . . . As im ( d ) = 0 then H ( G • ) = ker ( ∂ ) = C ( G ) .An interesting result is the following. Let T be aspanning tree of G . Note that if e ∈ G − T then T ∪ e has only one cycle: c e . The set of those cycles generate C ( G ) . Symbolically: < { c e ∀ e ∈ G − T } > = C ( G ) = H ( G • ) In other words: each spanning tree determines somebasis of H ( G • ) . In particular if T is a spanning treethat is a solution of the MSTCI problem then the treeintersection number (defined in the next section) of T could be a finer invariant of G .
2. Preliminaries
In the first part of this section we present some ofthe terms that are used in the article. Then we define thenotion of closest-point and closest-point-set . Finally weshow a convenient cycle partition.
Let G = ( V, E ) be a graph and T a spanning tree of G , then we will refer to the edges e ∈ T as tree-edges and to the edges e ∈ G − T as cycle-edges .Every cycle-edge e induces a cycle in T ∪ { e } , wewill call such a cycle a tree-cycle . And we shall call C T to the set of tree-cycles of T .The intersection of two tree-cycles is the set of edgesof T that belong to both cycles. We will define threefunctions concerning the intersection of tree-cycles.The first is ∩ T ( · , · ) : C T × C T → { , }∩ T ( c i , c j ) := ( c i ∩ c j = ∅ ∧ c i = c j c i ∩ c j = ∅ ∨ c i = c j Note that the trivial case c i = c j is excluded. Thisarbitrary decision will simplify future computations.The second is ∩ T ( · ) : C T → N ∩ T ( c i ) := X c j ∈ C T ∩ T ( c i , c j ) We will call ∩ T ( c ) the cycle intersection number of c . Given a tree-cycle c we will denote ∩ T,c as the set ofree-cycles that have non-empty intersection with c . Moreprecisely: ∩ T,c ≡ { c ′ ∈ C T : ∩ T ( c, c ′ ) = 1 } Note that | ∩
T,c | = ∩ T ( c ) .To define the third function consider T G to be the setof spanning trees of G , so the definition will be as follows: ∩ G : T G → N ∩ G ( T ) := 12 X c ∈ C T ∩ T ( c ) We will call ∩ G ( T ) the tree intersection number of T .Clearly the set min T ∩ G ( T ) is the set of solutions of theMSTCI problem. If the graph is clear from the context wecould drop the subindex and simply write: ∩ ( T ) .We shall call star spanning tree to a spanning tree thathas one vertex that connects to all other vertices. And K n to the complete graph on n nodes. If G = ( V, E ) we willsay that | V | = n is the number of vertices of G , | c | = k is the length of the cycle c and | p | is the length of thepath p . Also uT v will denote the unique path between u, v ∈ V in the spanning tree T ; d T ( v ) will be the degreeof v ∈ V relative to the spanning tree T . We will denote N ( v ) to the set of neighbor nodes of v ∈ V .Finally we use the terms “node” and “vertex” inter-changeably. In this section we prove the following simple fact: if G = ( V, E ) is a connected graph, T a spanning tree of G and c ∈ C T a tree-cycle, then for every node v ∈ V thereexists a unique node w ∈ c that minimizes the distance to v in T . We shall denote that node closest − point ( v, c ) . Lemma 1.
Let G = ( V, E ) be a connected graph, T aspanning tree of G and c ∈ C T a tree-cycle then forevery node v ∈ V there exists a unique node w ∈ c such that | vT w | ≤ | vT u | ∀ u ∈ c Proof. The proof proceeds by contradiction. If v ∈ c itis obviously its own unique closest point. Suppose that v / ∈ c and that there are two distinct nodes w, w ′ ∈ c such that | vT w | = | vT w ′ | ≤ | vT u | ∀ u ∈ c . Obviously w ′ / ∈ vT w and w / ∈ vT w ′ , we conclude that vT w ∪ wT w ′ ∪ vT w ′ determine a cycle in T which contradictsthe fact that T is a tree. (cid:3) The uniqueness of the closest − point ( v, c ) leads tothe following definition. Definition 2.
Let G = ( V, E ) be a connected graph, T a spanning tree of G and c ∈ C T a tree-cycle, thenthe set of closest points to a node w ∈ c is defined asfollows closest − point − set ( u, c ) := { v ∈ V − c : closest − point ( v, c ) = u } Now we define a partition of the set ∩ T,c . Moreprecisely, let G be a connected graph, T a spanning tree of G and c ∈ C T a tree-cycle. As defined above the set ∩ T,c is the set of tree-cycles that have non-empty intersectionwith c .Let us consider any tree-cycle c ′ ∈ ∩ T,c induced by acycle-edge e = ( v, w ) . In this setting we can define thefollowing partition: • Internal tree-cycles : c ′ is internal if v, w ∈ c . • External tree-cycles : c ′ is external if v / ∈ c and w ∈ c . • Transit tree-cycles : c ′ is transit if v, w / ∈ c .Let us denote them ∩ iT,c , ∩ eT,c , ∩ tT,c , respectively. Thispartition will be convenient to simplify the computationof the intersection number of c .
3. Tree cycles of complete graphs
In this section we analyze the complete graph case G = K n . First we deduce a formula to compute thecycle intersection number. Then we prove that the tree-cycles of a star spanning tree achieve the minimum cycleintersection number. Finally we conclude that the starspanning trees are the unique solutions of the MSTCIproblem. In this subsection we consider the problem of finding aformula to count tree-cycle intersections. More precisely,let G = K n , T a spanning tree of G and c a tree-cycle,we intend to derive a formula to calculate ∩ T ( c ) .The idea behind the formula is to consider the partitionof ∩ T,c , defined in the previous section. And then bycombinatorial arguments compute the number of elementsin each class.We shall analyze in turn the three classes: ∩ iT,c , ∩ eT,c , ∩ tT,c . In this section we will consider c ′ ∈ ∩ T ( c ) to be atree-cycle induced by a cycle-edge e = ( v, w ) .The simplest case is the internal tree-cycles class: ∩ iT,c . Let c ′ be an internal tree-cycle. By definition thenodes v and w belong to c , so the following holds: ( c ′ ∩ T ) ⊂ c because there is a unique path from v to w in T . So basically counting the number of internal tree-cycles reduces to count the pairings of the nodes of c excluding some obvious cases. The cases that should beexcluded are: the pairing of a node with itself and withits neighbors in c . Then the number of internal tree-cyclesis: | ∩ iT,c | = ( k − k , where k is, as before, equal to | c | . The quotient is obvi-ously due to the fact that every cycle is counted twice.ext we consider the class of external tree-cycles.Now let c ′ be an external tree-cycle. In this case ex-actly one of the extremal nodes ( v or w ) belong to c .Without loss of generality (as we are considering undi-rected edges), suppose that v / ∈ c and w ∈ c . Clearly w = closest − point ( v, c ) because in that case c ′ ∩ c = ∅ and consequently c ′ / ∈ ∩ T,c which contradicts our hy-pothesis. As that is the only particular case that should beexcluded, the number of external tree-cycles is: | ∩ eT,c | = ( n − k )( k − , where n = | V | is the number of vertices of G and k = | c | is the length of c .Last we consider the class of transit tree-cycles. Inthis case the key observation depends on the closest − point − set definition of the previous section. Lets definetwo classes of cycle-edges:1) A cycle-edge e = ( v, w ) is called intraset cycle-edge if both v, w ∈ closest − point − set ( u i , c ) for some u i ∈ c
2) A cycle-edge e = ( v, w ) is called interset cycle-edge if v ∈ closest − point − set ( u i , c ) and w ∈ closest − point − set ( u j , c ) where u i , u j ∈ c and u i = u j Then: • Every intraset cycle-edge induce a tree-cycle c ′ such that c ′ ∩ c = ∅ • Every interset cycle-edge induce a tree-cycle c ′ such that c ′ ∩ c = ∅ So we should consider interset cycle-edges or equiv-alently, the pairing of the nodes that are in different sets.Let q i = | closest − point − set ( w i , c ) | be defined for all w i ∈ c , then the number of transit tree-cycles is: | ∩ tT,c | = X i 1) + 12 k X i =1 q i ( n − k − q i ) , where n is the number of vertices of G , k = | c | and q i = | closest − point − set ( w i , c ) | for w i ∈ c . In this subsection we start by defining transiteless tree-cycles. Then we prove two lemmas. The first shows thatfor every cycle c ∈ G = K n we can build a spanningtree T such that c is a tree-cycle of T and the intersectionnumber ∩ T ( c ) is minimal. And the second calculates theintersection number of tree-cycles of star spanning trees.Finally we prove the main result of this section, namely that star spanning trees minimize ∩ ( · ) in the case ofcomplete graphs. Definition 3. Let G = ( V, E ) be a connected graph, T aspanning tree of G and c ∈ C T a tree-cycle, we call c a transitless tree-cycle if | ∩ tT,c | = 0 .As an important remark, note that the number ofelements in the internal and external classes of c areindependent of the spanning tree because they dependexclusively on the numbers n = | V | and k = | c | . Sotwo spanning trees T and T that have c as a tree-cycleinduce an intersection number (for c ) that only differ inthe number of elements in their transit classes. We con-clude that transitless tree-cycles have minimal intersectionnumber ∩ T ( c ) . Lemma 4. Let G = K n and let c be a cycle of G thenthe following procedure lead to a spanning tree T thatminimizes the intersection number of c :1) Remove exactly one edge e ∈ c 2) Choose a vertex v ∈ c 3) Define T as follows: • The edges c − e • A star centered at v of the vertices ( G − c ) ∪ { v } Proof. Note that T is a spanning tree of G , and c is atree-cycle of T . So if we prove that | ∩ tT,c | = 0 thenthe intersection number ∩ T ( c ) is minimal. This is thecase, because by construction: • | closest − point − set ( u, c ) | = 0 ∀ u ∈ c, u = v • | closest − point − set ( v, c ) | = n − k So | ∩ tT,c | = P i Let G = K n and let T s be a star spanning treeof G . Then the the following property holds ∩ T s ( c ) = 2( n − for any tree-cycle c of T s .Proof. Clearly the tree-cycles in T s have the sameintersection number (by symmetry). Let c be a tree-cycle of T s . Note that c is a triangle ( | c | = 3 ), sothe corresponding internal tree-cycles class is empty: | ∩ iT,c | = 0 . Also note that c is a transitless tree-cyclebecause its nodes are: the central node and two leafnodes of T s . So the external tree-cycle class is the onlynon-empty class: ∩ T s ( c ) = | ∩ eT s ,c | = 2( n − Proposition 6. Let G = K n and let T s be a star spanningtree of G . Then the following property holds ∩ T s ( · ) ≤ ∩ T ( · ) where T is any spanning tree of G .Proof. We shall prove the proposition by contradiction.Suppose that a spanning tree T and a tree-cycle c of T exist such that: ∩ T ( c ) < ∩ T s ( · ) = 2( n − e can assume that c is transitless because, if it’snot the case, by lemma 4 we can build a spanningtree T ′ such that ∩ T ′ ( c ) < ∩ T ( c ) . In this context theinequality can be expressed as ∩ T ( c ) = | ∩ iT,c | + | ∩ eT,c | =( k − k n − k )( k − < n − Expanding and simplifying the expression we have − k + ( n − 12 ) k − n + 6 < The roots of this quadratic polynomial are: r = 3 and r = 2( n − . We should consider two casesdepending on the relation of the roots:1) r < r r > r The case r = r can be discarded because it leadsto a fractional number of nodes ( n = ). In the firstcase the inequality holds for k < r = 3 or k > r =2( n − . The case k < is an obvious contradictionsince the size of the cycle must be | c | = k ≥ . Thecase k > n − combined with the fact that k ≤ n induces the following inequality r = 3 < r = 2( n − < k ≤ n which implies a contradiction: < n < , since n isa positive integer.The second case ( r = 3 > r = 2( n − ) imply n < . So the only case that should be considered is k = n = 3 since k ≤ n . But, this inequality is falsebecause k = 3 is a root of the quadratic polynomial. (cid:3) Corollary 7. Let G = K n and let T s be a star spanningtree of G . Then the the following property holds ∩ ( T s ) ≤ ∩ ( T ) , where T is any spanning tree of G .Proof. As expressed by proposition 6, a tree-cycle of astar spanning tree has the minimal intersection numberamong all tree-cycles. Since any tree-cycles of a starspanning tree has the same intersection number, weconclude that the tree intersection number of a starspanning tree ∩ ( T s ) is minimal among all spanningtrees. (cid:3) This corollary can be further improved to a strictinequality. In other words: star spanning trees are theunique minimizers of ∩ ( T ) . Corollary 8. Let G = ( V, E ) = K n where | V | = n > and let T s be a star spanning tree of G . Then, thefollowing property holds ∩ ( T s ) < ∩ ( T ) , where T is any non-star spanning tree of G .Proof. A careful reading of proposition 6 leads tothe conclusion that the equality ∩ T s ( c ) = ∩ T ( c ) isachieved when k is either r = 3 or r = 2( n − (the roots of the quadratic polynomial). If k = r = 2( n − , and taking into account that ≤ k ≤ n ,we conclude that ≤ n ≤ ; this case is explicitlyexcluded from our hypotheses (in fact, it is not difficultto check that the three non-isomorphic spanning treesof K all have the same tree intersection number).The other possibility is k = r = 3 . As all the tree-cycles of T s fall into this category, it is enough toshow that T has a tree-cycle c such that | c | = k > to conclude our thesis. Let w ∈ V be a node withmaximum degree in T . And let d T ( w ) denote thedegree of w in T and N ( w ) to the set of neighborsof w in T . As T is a non-star spanning tree then ≤ d T ( w ) < n − . So there is a node u ∈ V such that u / ∈ N ( w ) in T , and there is a node k ∈ N ( w ) suchthat k / ∈ wT u . Notice that the edge e = ( u, k ) / ∈ T (since it would induce a cycle). Hence, it is a cycle-edge. Note that the tree-cycle induced by e has lengthat least 4. (cid:3) This result can be summarized in the following way:star spanning trees are the unique solutions of the MSTCIproblem for complete graphs. 4. Further generalization Now we explore some aspects of a slightly moregeneral case, namely: the MSTCI problem in the contextof a graph (not necessarily complete) G = ( V, E ) thatadmits a star spanning tree T s . In the first part we presenta formula to calculate ∩ ( T s ) . In the second part we showthat ∩ ( T s ) is a local minimum in the domain of what werefer to as the “spanning tree graph”. In the third part weprove a result that suggests a general observation: the factthat a spanning tree of a graph G being a solution of theMSTCI problem doesn’t depend on an intrinsic propertyof T but on the particular embedding of T in G . Finally weconjecture a generalization of Corollary 7: ∩ ( T s ) ≤ ∩ ( T ) for every spanning tree T of G . In this subsection we present two formulas for graphs G = ( V, E ) that admit a star spanning tree T s . Let usdenote v ∈ V to the central node of T s .The first formula corresponds to the cycle intersectionnumber of a tree-cycle c = ( u, v, w ) ∈ C T s , namely ∩ T s ( c ) . Recall from the previous section that c does notintersect neither transit nor internal tree-cycles: |∩ tT,c | = 0 and |∩ iT,c | = 0 . So its non-empty intersections are the tree-cycles in the set ∩ eT,c . Note that the remaining incidentedges to u and w , are the only source of tree-cycles thathave non-empty intersection with c . So the formula isstraightforward: ∩ T s ( c ) = d ( u ) − d ( w ) − , where d ( u ) and d ( w ) are the degrees of u and w , resp.Now we shall deduce a formula for the tree intersec-tion number ∩ ( T s ) . Based on the preceding observationsand the fact that each node u ∈ V − { v } is involved in ( u ) − tree-cycles and for each of those tree-cycles itproduces d ( u ) − non-empty intersections (note that thepairwise intersections are counted twice). The formula isas follows: ∩ ( T s ) = 12 X u ∈ V −{ v } ( d ( u ) − d ( u ) − 2) =12 X u ∈ V −{ v } d ( u ) − d ( u ) + 2 If we denote d as the degree vector of G , that is: avector that has in the i -th component the degree of the i -th vertex. And taking into account that P u ∈ V d ( u ) = 2 m where m = | E | , the formula can be expressed as: ∩ ( T s ) = 12 [ || d || − m − ( n − n − In this subsection we prove that a star spanning tree isa local minimum respect to the tree intersection numberin the domain of the spanning tree graph . We start bydefining this second order graph of the original graph G =( V, E ) . Then we analyze the structure of the neighbors ofa star spanning tree T s . Finally we demonstrate the resultby a bijection between tree-cycles to conclude that ∩ ( T s ) is a local minimum. Definition 9. Let G = ( V, E ) be a graph, and S asubgraph of G . We denote as e ↔ e ′ to the operationof replacing the edge e ∈ S with the edge e ′ ∈ G − S .We call this operation edge replacement on S . Definition 10. Let G = ( V, E ) be a graph. We denote SP G to the graph that has one node for every spanningtree of G and an edge between two nodes if thecorresponding spanning trees differ in exactly oneedge replacement. We call this graph the spanning treegraph of G . T s . Let G = ( V, E ) be a graphthat admits a star spanning tree T s with v ∈ V as itscenter. Let α Ts be the node corresponding to T s in SP G and let α T (with corresponding spanning tree T ) be anyneighbor of α Ts . By definition T s and T differ in exactlyone edge replacement e ↔ e ′ where e = ( v, w ) ∈ T s and e ′ = ( u, w ) ∈ T . Note that T is exactly the same as T s except that the node w is no longer connected to thecentral node v but is connected to the intermediate node u . This similar structure has direct consequences in theintersection numbers of both trees.Now we prove the result of this section. Theorem 11. Let G = ( V, E ) be a graph that admits astar spanning tree T s with v ∈ V as its center. Then, T s is a local minimum respect to the tree intersectionnumber in the domain of SP G .Proof. Let T be a spanning tree corresponding toa neighbor of T s in SP G . Then we want to provethat ∩ ( T s ) ≤ ∩ ( T ) . We shall proceed by defining a bijection between the tree-cycles of both trees { c ↔ d : c ∈ C Ts ∧ d ∈ C T } such that ∩ Ts ( c ) ≤ ∩ T ( d ) , thisstrategy clearly implies the thesis since by definition: ∩ ( T s ) = 12 X c ∩ T s ( c ) ≤ X d ∩ T ( d ) = ∩ ( T ) Let e Ts ↔ e T with e Ts = ( v, w ) ∈ T s and e T =( u, w ) ∈ T be the edge replacement in SP G . Considerthe following simple facts: • e T is a cycle-edge in T s , with corresponding tree-cycle c • e Ts is a cycle-edge in T , with corresponding tree-cycle d • Except for e Ts and e T , T s and T have the sameset of cycle-edges. For every e ∈ E − T s − T wedenote c e and d e to the corresponding tree-cyclesin T s and T , resp.according to this naming convention, we can definethe following “natural” bijection between tree-cyles: { c ↔ d } ∪ { c e ↔ d e : e ∈ E − T s − T } In order to compare the intersection numbers of the bi-jected pairs it is convenient to distiguish the followingpartition: • Case 1: the pair induced by the edge replacement, { c ↔ d } • Case 2: pairs induced by cycle-edges non-incidentto u nor to w , { c e ↔ d e : e ∈ E − T s − T ∧ u / ∈ e ∧ w / ∈ e } • Case 3: pairs induced by cycle-edges incident to u or w , { c e ↔ d e : e ∈ E − T s − T ∧ ( u ∈ e ∨ w ∈ e ) } Case 1 is the easiest: note that c and d are the sametree-cycle ( u, v, w ) , which is a transitless triangle, soits intersection number is determined by its externalintersections: ∩ Ts ( c ) = d ( u ) − d ( w ) − ∩ T ( d ) Case 2 is similar, let e = ( h, k ) be a cycle-edge non-incident to u or to w and c e ↔ d e its correspondingpair of bijected tree-cycles. Clearly e determines thetransitless triangle ( h, v, k ) both in T s and T and as d Ts ( h ) = d Ts ( k ) = d T ( h ) = d T ( k ) = 1 , then everyother edge incident to h or k induces a tree-cycle thatintersects ( h, v, k ) . We conclude that: ∩ Ts ( c e ) = d ( h ) − d ( k ) − ∩ T ( d e ) Case 3 is the one that should be analyzed more care-fully. As we already know how to calculate intersec-tion numbers of tree-cycles in T s , we will focus on thetree-cycles of T . We will further divide this partitionin two subpartitions: • Case 3.1: pairs induced by cycle-edges incident to u , { c e ↔ d e : e ∈ E − T s − T ∧ u ∈ e } • Case 3.2: pairs induced by cycle-edges incident to w , { c e ↔ d e : e ∈ E − T s − T ∧ w ∈ e } n case 3.1 the situation is as follows: the cycle-edge e = ( u, k ) defines the tree-cycle c e = d e = ( u, v, k ) (both in T and T s ). The important details are: • d T ( u ) = 2 : u induces d ( u ) − intersections • d T ( k ) = 1 : k induces d ( k ) − intersections • d T ( w ) = 1 : w induces d ( w ) − intersections • d ( w ) ≥ since it is connected at least to u and v in G • w may have an incident cycle-edge connectingit to k , so we should avoid counting twice thatintersectionNow we claim that ∩ T ( d e ) ≥ d ( u ) − d ( k ) − d ( w ) − − ǫ ( w, k ) ≥ d ( u ) − d ( k ) − ∩ Ts ( c e ) where ǫ ( w, k ) = ( w, k ) ∈ E otherwise The inequality follows since d ( w ) − − ǫ ( w, k ) ≥ .In case 3.2 the situation is as follows: the cycle-edge e = ( w, h ) defines the tree-cycle d e = ( w, u, v, h ) in T and c e = ( w, v, h ) in T s . The important details are: • d T ( u ) = 2 : u induces d ( u ) − intersections • d T ( h ) = 1 : h induces d ( h ) − intersections • d T ( w ) = 1 : w induces d ( w ) − intersections • u may have an incident cycle-edge connectingit to h , so we should avoid counting twice thatintersectionAnd we claim that ∩ T ( d e ) ≥ d ( w ) − d ( h ) − d ( u ) − − ǫ ( u, h ) ≥ d ( w ) − d ( h ) − ∩ Ts ( c e ) The inequality follows since d ( u ) − − ǫ ( u, h ) ≥ . (cid:3) In this subsection we consider the following question:is there any correlation between an intrinsic tree invariant and the tree intersection number of the spanning treesfor every graph? If so we could formulate an alternativecharacterization of the MSTCI problem expressed in termsof the invariant.By intrinsic tree invariant we denote a map f : T → R on the set of all trees. Of particular interest arethe degree-based topological indices [Gutman, 2013]. Thetopological index that motivated our question is the atom-bond connectivity (ABC) index [Estrada, 1998]. As shownby [Furtula, 2009] the star trees are maximal among alltrees respect to the ABC index. In the previous sectionwe proved that in the complete graph the star spanningtrees are minimal respect to the tree intersection number.Consequently we can formulate a natural question: is there a negative correlation between the ABC index ofthe spanning trees and their corresponding intersectionnumbers?We will prove that the answer to our question isnegative. Without loss of generality we will considerpositive correlation (negative correlation is analogous).The underlying idea of the proof is as follows: supposethat there exists an intrinsic tree invariant f : T → R such that for every graph G the intersection number ∩ ( · ) is positively correlated with f , this can be expressed as: f ( T ) ≤ f ( T ) ⇐⇒ ∩ G ( T ) ≤ ∩ G ( T ) , ∀ G, T , T According to this property if we consider two trees T and T and two graphs G and H such that T , T ∈ T G and T , T ∈ T H , then this equivalence follows: ∩ G ( T ) ≤ ∩ G ( T ) ⇐⇒ ∩ H ( T ) ≤ ∩ H ( T ) So it suffices to show that there exist T , T , G and H such that the equivalence is not satisfied to answer thequestion negatively.First we prove a simple lemma regarding the tree in-tersection number of a spanning tree T under the removalof a cycle-edge. Namely if a cycle-edge e is removed from G then the tree intersection number of T decreases exactlyin the intersection number of its corresponding tree-cycle. Lemma 12. let G = ( V, E ) be a graph, T ∈ T G aspanning tree, e ∈ G − T a cycle-edge, and c thecorresponding tree-cycle, then the following holds: ∩ G − e ( T ) = ∩ G ( T ) − ∩ T ( c ) Proof. As the spanning tree T is the same in bothgraphs: G and G − e , then the remaining cycle-edgesdefine the same tree-cycles so their pairwise intersec-tion relations are identical. As c is not a cycle in G − e then the equality follows. (cid:3) Theorem 13. There is no intrinsic tree invariant f : T → R positively correlated with the intersection number ∩ G ( · ) for every graph G .Proof. We will proceed by contradiction: let f besuch an intrinsic tree invariant. Then by definition forarbitrary graphs G and H the following equivalenceshold f ( T ) ≤ f ( T ) ⇐⇒ ∩ G ( T ) ≤ ∩ G ( T ) f ( T ) ≤ f ( T ) ⇐⇒ ∩ H ( T ) ≤ ∩ H ( T ) Where T , T ∈ T G and T , T ∈ T H . This in turnimply that ∩ G ( T ) ≤ ∩ G ( T ) ⇐⇒ ∩ H ( T ) ≤ ∩ H ( T ) The proof will be based on showing two graphs andtwo spanning trees such that the latter equivalence isnot valid. • Let G be the complete graph K n Let H be the graph K n − { e i, , . . . , e i,n − } wherethe edges e i, , . . . , e i,n − are n − edges incidentto some arbitrary node v i . We will refer to v i as the almost disconnected node of H . Note that d ( v i ) =2 . • Let T be the star spanning tree T s • Let T be the spanning tree defined as T s − { e i } ∪{ e i,j } , where e i is the edge that connects somearbitrary node v i (in H this role will be playedby the almost disconnected node) to the center ofthe star and e i,j is an edge that connects v i to adifferent node v j .It is easy to check that T and T are spanning trees ofboth G and H . If we also suppose that | V | = n > then by corollary 8 ∩ G ( T ) < ∩ G ( T ) By the previous equivalence it is expected that ∩ H ( T ) < ∩ H ( T ) as well. But we will show thatthis is not the case.By a suitable labelling of the nodes of H we can referto: the center of the star spanning tree as v , the almostdisconnected node of H as v and the other neighborof v as v . By lemma 12 arises that ∩ H ( T ) = ∩ H − e , ( T ) + ∩ T ( c , ) ∩ H ( T ) = ∩ H − e , ( T ) + ∩ T ( c , ) Where c , and c , are the tree-cycles induced by e , and e , in T and T , resp. Since the remaining tree-cycles corresponding to both trees are the same then ∩ H − e , ( T ) = ∩ H − e , ( T ) And this imply the following ∩ H ( T ) − ∩ H ( T ) = ∩ T ( c , ) − ∩ T ( c , ) It is an easy exercise to check that ∩ T ( c , ) = ∩ T ( c , ) = d ( v ) − n − At this point we can conclude that ∩ H ( T ) = ∩ H ( T ) Contradicting the fact that f is positively correlatedwith the tree intersection number for every graph. (cid:3) The underlying key fact of this result is that a spanningtree T that solves the the MSTCI problem for a graph G does not depend on intrinsic properties of T but on theembedding of T in G .Note that as an interesting side effect this demonstra-tion shows that a star spanning tree is not necessarilya strict local minimum in the spanning tree graph (seeprevious subsection). In this subsection we present the conjecture ∩ ( T s ) ≤∩ ( T ) for every spanning tree T generalizing theorem 11.Then we explore two ideas to simplify a hypotheticalcounterexample of the conjecture. The first is based onthe notion of interbranch cycle-edge. We show that if anon-star spanning tree T exists such that ∩ ( T ) < ∩ ( T s ) ,then the inequality must hold if we remove the interbranchcycle-edges. The second is based on the notion of princi-pal subtree . In this case we show that the inequality musthold for some principal subtree of T . This ideas will beof practical use in the next section. We present below theconjecture that generalizes the case of complete graphs. Conjecture 14. Let G = ( V, E ) be a graph that admits astar spanning tree T s , then ∩ ( T s ) ≤ ∩ ( T ) for every spanning tree T ∈ T G .As an important remark, a demonstration of this resultseems difficult if approached by a local-to-global strategyas in the complete graph case exposed previously. In this part weconsider some ideas to simplify a hypothetical counterex-ample of conjecture 14.Below we define the notion of interbranch cycle-edge. Definition 15. Let G = ( V, E ) be a graph that admits astar spanning tree T s and let v ∈ V be the center of T s .Let T ∈ T G be a spanning tree. We call interbranchcycle-edge of T to any cycle-edge of T , e = ( u, w ) ,such that closest − point ( v, c ) = u, w , where c is theinduced tree-cycle of e in T .The intuition behind this definition is that the paths vT u and vT w belong to different branches. Or equiv-alently, u and w are not collinear with respect to v in T . The following lemma shows that if we can find acounterexample to the conjecture 14 (ie.: ∩ ( T ) < ∩ ( T s ) )then we can build a simpler counterexample removingfrom G the interbranch cycle-edges of T . Lemma 16. Let G = ( V, E ) be a graph that admits a starspanning tree T s with v ∈ V as its center. Let T ∈ T G be a spanning tree such that ∩ G ( T ) < ∩ G ( T s ) and let ∆ T be the set of interbranch cycle-edges of T , then ∩ G − ∆ T ( T ) < ∩ G − ∆ T ( T s ) Proof. Let e = ( u, w ) ∈ ∆ T . Note that e is also acycle-edge in T s since v = u, w by definition of inter-branch cycle-edge. So e determines the tree-cycle c in T s and the tree-cycle c ′ in T . By the intersection num-ber formula it arises that ∩ Ts ( c ) = d ( u ) − d ( w ) − .On the other hand, since the other neighbors of u and are connected to v , they belong to distinct tree-cycles with non-trivial intersection with respect to c ′ in T . We conclude that ∩ T ( c ′ ) ≥ d ( u ) − d ( w ) − ∩ T s ( c ) . Hence by lemma 12, ∩ G − e ( T ) = ∩ G ( T ) − ∩ T ( c ′ ) < ∩ G ( T s ) − ∩ T s ( c ) = ∩ G − e ( T s ) Applying the same procedure for every edge in ∆ T ,the claimed inequality follows. (cid:3) Definition 17. Let T = ( V, E ) be a rooted tree graph withroot v ∈ V . Let w ∈ N ( v ) then we call principalsubtree respect to w to the subtree spanned by v andthe nodes u ∈ V such that w ∈ vT u .The next lemma expresses the intersection number ofa spanning tree (without interbranch cycle-edges) as thesum of the intersection number of its principal subtrees. Lemma 18. Let G = ( V, E ) be a graph that admits a starspanning tree T s with v ∈ V as its center. Let T bea spanning tree of G without interbranch cycle-edges(ie: ∆ T = ∅ ), then the following holds ∩ G ( T ) = X w ∈ N ( v ) ∩ G w ( T w ) , where T w is the principal subtree of w ∈ N ( v ) considering T as a rooted tree with v as its root. And G w is the subgraph spanned by T w .Proof. As ∆ T = ∅ there are no cycle-edges con-necting any two such principal subtrees. This impliesthat the nonempty intersections between tree-cycles of T must occur inside each subtree. This determines apartition of C T and the claimed expression follows. (cid:3) The following corollary in line with lemma 16 furthersimplifies a hypothetical counterexample of conjecture 14. Corollary 19. Let G = ( V, E ) be a graph that admits a starspanning tree T s with v ∈ V as its center. Let T bea spanning tree of G without interbranch cycle-edges(ie: ∆ T = ∅ ) such that ∩ ( T ) < ∩ ( T s ) then ∩ ( T w ) < ∩ ( G w ∧ T s ) for some G w . Where T w is the principal subtree of w ∈ N ( v ) considering T as a rooted tree with v asits root; G w is the subgraph of G spanned by T w ; G w ∧ T s is the subtree of T s restricted to G w , namelythe intersection between G w and T s .Proof. First note that the G w ’s are edge disjointsince ∆ T = ∅ . This partition of the edges of G also determines a partition of T s such that ∩ ( T s ) = P w ∈ N ( v ) ∩ ( G w ∧ T s ) . As the parts are in a naturalbijective relation since they are the subtrees of T and T s restricted to each G w , we can express theintersection number of T and T s as follows ∩ ( T ) = X w ∈ N ( v ) ∩ ( T w ) < X w ∈ N ( v ) ∩ ( G w ∧ T s ) = ∩ ( T s ) And from the bijection we can deduce that ∩ ( T w ) < ∩ ( G w ∧ T s ) for some G w . (cid:3) 5. Programmatic exploration In this section we present some experimental resultsto reinforce conjecture 14. We proceed by trying to find acounterexample based on our preceding observations. Inthe first part we focus on the complete analysis of smallgraphs, ie: graphs of at most 9 nodes. In the second partwe analyze larger families of graphs by random samplinginstances. In the previous section we showed that the spaceof candidate counterexamples of conjecture 14 can bereduced. The general picture is as follows: • Let G = ( V, E ) be a graph that admits a starspanning tree T s with v ∈ V as its center • In the case that we can find some non-star span-ning tree T of G such that ∩ ( T ) < ∩ ( T s ) • Then we can “simplify” the instance by removingthe interbranch cycle-edges with respect to T in G without affecting the inequality (see lemma 16) • We can further reduce the instance by focusing onthe case where d T ( v ) = 1 , that is: the degree of v restricted to T is 1 (see corollary 19)This considerations can be used to implement al-gorithms to explore the space of spanning trees moreefficiently. Since the algorithms will generate instancesin this ‘reduced’ form instead of a brute force approach. In this subsection we present an algorithm to explorethe spanning tree space. The algorithm proceed by exhaus-tively analyzing all the reduced graphs of a given numberof nodes. The size of the space increases exponentiallywith respect to the number of nodes, so it has a majorlimitation: it can be used to analyze only small graphs.The main part is sketched in Algorithm 1.The details of the algorithm are the following: • The input parameter n is the number of nodes ofthe graphs to explore • GenerateAllT rees ( n − is a function that re-turns the list of all trees of n − nodes • GenerateGraph ( w, T ′ ) is a function that buildsa graph G . Based on the tree T ′ , it adds a newnode ( v ) that will play the role of the central nodeof a star spanning tree, then adds the edge ( v, w ) to define our candidate tree counterexample T .Finally adds all the other edges that link v to therest of the nodes to obtain G . It returns the graph G and ( ¯∆ ) the set of “possible” non-interbranchcycle edges. • IntersectionN umber ( φ, G ) is a function thatcalculates the intersection number of T in G ∪ φ ,where φ ⊂ ¯∆ is a subset of supplementary edgesof G . able 1. S MALL INSTANCES RESULTS nodes instances (approx.)4 55 336 2517 42008 1250009 7900000 • StarIntersectionF ormula ( φ, G ) is a functionthat calculates the intersection number of the starspanning tree in G ∪ φ • The algorithm finds a counterexample of theconjecture if: IntersectionN umber ( φ, G ) CounterexampleSearch( n ) T ← GenerateAllT rees ( n − for each tree T ′ ∈ T dofor each node w ∈ T ′ do G, ¯∆ ← GenerateGraph ( w, T ′ ) for each subset φ ⊂ ¯∆ docheck ( IntersectionN umber ( φ, G ) ANDOM INSTANCES RESULTS nodes instances25 300000050 300000100 30000200 15000400 300 the edges that link v to the rest of the nodes toobtain G . It returns the graph G and ( ¯∆ ) a randomset of non-interbranch cycle edges. • IntersectionN umber ( φ, G ) same as algorithm 1 • StarIntersectionF ormula ( φ, G ) same as algo-rithm 1 • The algorithm finds a counterexample of theconjecture if: IntersectionN umber ( φ, G ) CounterexampleRandomSearch( n, k ) for i := 1..k do T ← GenerateRandomT ree ( n ) G, ¯∆ ← GenerateRandomGraph ( T ) check ( IntersectionN umber ( φ, G ) Bernoulli trial to each such possible edge.To achieve some diversity for each tree we built threedifferent sets to obtain a sparse, medium and dense setsbased on corresponding probabilities . , . , . .The proposed algorithm did not find a counterexampleof the intersection conjecture. Table 2 shows the size ofthe experiments. The column nodes is the number of nodesof the graph family, ie: | V | . The column instances is thenumber of instances processed. 6. Conclusion In this article we introduced the Minimum SpanningTree Cycle Intersection (MSTCI) problem.We proved by enumerative arguments that the starspanning trees are the unique solutions of the problemin the context of complete graphs.We conjectured a generalization to the case of graphs(not necessarily complete) that admit a star spanning tree.In this sense we showed that the star spanning tree is alocal minimum in the domain of the spanning tree graph .We deduced a closed formula for the tree intersectionnumber of star spanning trees in this setting. We proposedtwo ideas to attempt to find a counterexample of theconjecture. Those ideas were the basis of two strategiesto programatically explore the space of solutions in theursue of a counterexample. The negative result of theexperiments suggest that the conjecture is well posed.Unlike the complete graph context, in this slightly moregeneral case, star spanning trees are not unique; there areother spanning trees T such that ∩ ( T s ) = ∩ ( T ) .We proved a general result that shows that spanningtrees that solve the MSTCI problem don’t depend on someintrinsic property but on their particular embedding in theambient graph.An interesting direction of research is to consider theMSTCI problem for other families of graphs, ie.: graphsthat do not admit a star spanning tree. Of particular interestis the class of triangular meshes, ie.: graphs that modelthe immersion of compact surfaces in the 3D euclideanspace.Another interesing direction of research is related toproving to which complexity class the MSTCI problembelongs to. In case of belonging to the NP-hard class, itwill be important to find approximate, probabilistic andheuristic algorithms. References [Botsch, 2010] Botsch, M. and Kobbelt, L. and Pauly, M. and Alliez,P. and L´evy, B., Polygon Mesh Processing , A. K. Peters, 2010.[Cartan, 1956] Cartan, H. P. and Eilenberg, S., Homological Algebra ,Princeton University Press, 1956.[Estrada, 1998] Estrada E. and Torres L. and Rodr´ıguez L. and GutmanI., An atom-bond connectivity index: modelling the enthalpy offormation of alkanes , Indian J. Chem. 37A (1998) 849–855.[Furtula, 2009] Furtula B. and Graovac A. and Vukiˇcevi´c D., Atom-bond connectivity index of trees , Discrete Appl. Math. 157 (2009)2828–2835.[Gutman, 2013] Gutman I., Degree-based topological indices , Croat.Chem. Acta 86 (2013) 351–361.[Dewan, 2010] Dewan I., Graph Homology and Cohomology .[McKay and Piperno, 2014] McKay B.D. and Piperno A., PracticalGraph Isomorphism, II , Journal of Symbolic Computation, 60(2014), pp. 94-112.[McKee and McMorris, 1999] McKee T.A and McMorris F.R., Topicsin intersection graph theory , SIAM monographs on discrete math-ematics and applications, 1999.[Tutte, 1965] Tutte W.T.,