aa r X i v : . [ m a t h . C O ] M a y SPLITTERS AND DECOMPOSERS FOR BINARY MATROIDS
S. R. Kingan Department of MathematicsBrooklyn College, City University of New YorkBrooklyn, NY [email protected]
Abstract.
Let EX [ M . . . , M k ] denote the class of binary matroids with no minors isomorphicto M , . . . , M k . In this paper we give a decomposition theorem for EX [ S , S ∗ ], where S isa certain 10-element rank-4 matroid. As corollaries we obtain decomposition theorems for theclasses obtained by excluding the Kuratowski graphs EX [ M ( K , ) , M ∗ ( K , ) , M ( K ) , M ∗ ( K )]and EX [ M ( K , ) , M ∗ ( K , )]. These decomposition theorems imply results on internally 4-connectedmatroids by Zhou [8], Qin and Zhou [7], and Mayhew, Royle and Whittle [4]. Introduction
Some matroids such as the complete graph on five vertices K , the complete bipartite graph withthree vertices in each class K , , and the Fano matroid F play a more prominent role in structuretheory than others. Let EX [ M . . . , M k ] denote the class of binary matroids with no minorsisomorphic to M , . . . , M k . In 1937 Wagner building on Kuratowski’s work identified the minimalexcluded minors for the class of planar graphs as K and K , [6, 5.2.5]. In 1958 Tutte identified theminimal excluded minors for the class of regular matroids as F and its dual F ∗ [6, 10.1.2]. In 1980Seymour gave a decomposition result for the class of regular matroids. In his decomposition result,we meet two matroids R and R that play a central role for regular matroids. The matroid R is a 10-element rank-5 self-dual matroid. It is a splitter for 3-connected regular matroids. Thematroid R is a 12-element rank-6 self-dual matroid. It is a 3-decomposer for 3-connected regularmatroids. Seymour proved that if M is a 3-connected matroid in EX [ F , F ∗ ], then either R is a3-decomposer for M or M is a graph, cograph, or R [6, 13.1.2]. We talk of 3-connected matroidsbecause matroids that are not 3-connected can be decomposed into their 3-connected componentsby direct sums or 2-sums [6, 8.3.1].From a structural point of view, the regular matroids are taken care of and the focus shifts tomatroids that have F or F ∗ as a minor. Observe that F is the binary projective plane P G (2 , F and itsdual F ∗ are given below. By a slight abuse of notation, we will treat a vector matroid M [ A ] assynonymous with the matrix A , when the context and representation is clear. F = I F ∗ = I The 3-connected single-element extensions of F ∗ are AG (3 ,
2) and S , both of which are self-dual,so they are also the 3-connected coextensions of F . The author is partially supported by PSC-CUNY grant number 66305-00 44. AG (3 ,
2) = I S = I S has two 3-connected binary non-isomorphic single-element extensions P and Z and AG (3 , Z . P has three single-element extensions, one of whichis the internally 4-connected matroid S . In [5] Oxley flagged P as important by obtaining acomplete characterization of the 3-connected binary non-regular matroids with no minor isomor-phic to P or P ∗ . This made the new starting point for investigating binary matroids P or P ∗ .Observe that P has a non-minimal exact 3-separation ( A, B ), where A = { , , , } is both acircuit and a cocircuit. Matrix representations for P and S are given below: P = I S = I In this paper we highlight S and give a decomposition theorem for EX [ S , S ∗ ]. In the processwe flag two of the single-element coextensions of P known as E and E as significant. Matrixrepresentations are shown below: E = I E = I The matroids E is internally 4-connected, whereas E is not internally 4-connected. All thesematroids play signficiant roles in the characterization of the almost-graphic matroids and thealmost-regular matroids with at least one regular element [2]. The matroid S is the first matroidin the internally 4-connected infinite family of almost-graphic matroids S n +1 . The almost-regularmatroids with at least one regular element are precisely the almost-regular matroids with no E -minor. Finally, the matroid T also makes an appearance in our characterization. It is a12-element rank-6 self-dual 4-connected matroid [ ?? ]. A matrix representation is given below. T = I The next two theorems are the main results of this paper. In Theorem 1.1 we give a setof splitters and 3-decomposers that together characterize the excluded minor class EX [ S , S ∗ ].Unlike the class of regular matroids that has just one 3-decomposer R whose non-minimal exact 3-separation is induced in all 3-connected regular matroids containing it, EX [ S , S ∗ ] has several 3-decomposers with different inducers. In fact, E has two different non-minimal exact 3-separationsand it is not possible to determine conclusively which of the two gets induced in the 3-connectedmatroids with an E -minor. We only show that at least one of the two 3-separations must be aninducer. Thus there is no way of determining how the 3-connected matroids in EX [ S , S ∗ ] are PLITTERS AND DECOMPOSERS FOR BINARY MATROIDS 3 pieced together across 3-sums, which is something Seymour was able to do for regular matroids.Nonetheless, when the focus shifts from a particular non-minimal exact 3-separation to a set of3-decomposers with different non-minimal exact 3-separations, we get a decomposition theoremwith a relatively short proof. Several recent theorems on internally 4-connected matroids followimmediately.
Theorem 1.1.
Let M be a -connected non-regular matroid in EX [ S , S ∗ ] . Then S , P , P ∗ ,or E (with either of two inducers) is a -decomposer for M or M is isomorphic to F , F ∗ , E , T \ e , T /e ,or T . Once we establish that a matroid in the class has a non-minimal exact 3-separation inducedby one of the listed 3-decomposers, we are not concerned with it any more. We know it can bedecomposed into smaller parts. It is the other ones that won’t admit a decomposition that are ofinterest. The rank 3 extremal matroid (monarch) is F and the rank 6 extremal matroid is T .The rank 4 and 5 extremal matroids are not important because they admit non-minimal exact3-separation.Next, observe that S is the only single-element extension of M ∗ ( K , ) and E is a non-regularsingle-element coextensions of M ∗ ( K , ). A decomposition result for EX [ M ( K , ) , M ∗ ( K , )] fol-lows directly from Theorem 1.1 because by excluding S or S ∗ we are effectively excluding M ( K , ) or M ∗ ( K , ). No additional work has to be done to get the next major new decomposi-tion theorem; just remove E from Theorem 1.1. Theorem 1.2.
Let M be a -connected binary non-regular matroid in EX [ M ( K , ) , M ∗ ( K , )] .Then S , P , P ∗ , or E (with either of two inducers) is a -decomposer for M or M is isomorphicto F , F ∗ , T \ e , T /e ,or T . When the extremal matroids are known all the internally 4-connected matroids are easily foundbecause they are the minors of the extremal matroids. The main proofs of three recent resultsfollow immediately as corollaries. The first corollary is the main theorem in [8, Theorem 1.1]. Thereader will want to know that Zhou had different names for the matroids. In his paper E = P , E = N , and S = ˜ K . Our notation of E and E is from the almost-graphic paper and adozen other papers. So we want to stick with it for consistency. Additionally, Zhou also talkedabout a single-element coextension of S called Q (which we call E ), but we show in the proofof the main theorem that it is not significant because it is decomposed by P . Corollary 1.3. (Zhou 2004) Let M be an internally -connected binary non-regular matroid in EX [ S , S ∗ ] . Then M is isomorphic to F , F ∗ , E , T , T \ e , or T /e . (cid:3) We already know from [1] that EX [ M ( K , ) , M ∗ ( K , ) , M ( K ) , M ∗ ( K )] is mostly the same as EX [ M ( K , ) , M ∗ ( K , )]. The main theorem in [1, Theorem 2.1] states that, if M is a 3-connectedbinary matroid with an M ( K )-minor, then either M has an M ( K , )- or M ∗ ( K , )-minor or M isisomorphic to M ( K ), T /e , or T . Moreover, T is a splitter for EX [ M ( K , ) , M ∗ ( K , )]. It fol-lows that matroids in EX [ M ( K , ) , M ∗ ( K , )], but not in EX [ M ( K , ) , M ∗ ( K , ) , M ( K ) , M ∗ ( K )]are precisely M ( K ), M ∗ ( K ), T \ e , T /e , or T .Mayhew, Royle, and Whittle’s identification of the internally 4-connected matroids in EX [ M ( K , ) , M ∗ ( K , )] in [4] follows directly from Theorem 1.3. So does the main result ofQin and Zhou’s paper where they exclude the Kuratowski graphs and their duals from binarymatroids [7, Theorem 1.3]. Note that if a matroid M in EX [ M ( K , ) , M ∗ ( K , ) , M ( K ) , M ∗ ( K )]is regular, then M is isomorphic to a planar graph. If a matroid in EX [ M ( K , ) , M ∗ ( K , )] isregular, then M is isomorphic to a planar graph or K or K ∗ . SPLITTERS AND DECOMPOSERS FOR BINARY MATROIDS
Corollary 1.4. (Mayhew, Royle, Whittle) Let M be an internally -connected binary non-regularmatroid in EX [ K , , K ∗ , ] . Then M is isomorphic to F , F ∗ , T , T \ e , or T /e . (cid:3) Corollary 1.5. (Qin and Zhou 2004) Let M be an internally -connected binary non-regularmatroid in EX [ M ( K , ) , M ∗ ( K , ) , M ( K ) , M ∗ ( K )] . Then M is isomorphic to F or F ∗ . (cid:3) In Section 2 we explain the terminology and techniques used in the paper. In Section 3 we givethe proof of Theorem 1.1. Note that Theorem 1.2 and the corollaries are immediate consequencesof Theorem 1.1. 2.
The techniques used in the main theorem
The matroid terminology follows Oxley [6]. If M and N are matroids on the sets E and E ∪ e where e E , then M is a single-element extension of N if M \ e = N , and M is a single-elementcoextension of N if M ∗ is a single-element extension of N ∗ . Let M be a class of matroids closedunder minors and isomorphism. A splitter N for M is a 3-connected matroid in M such thatno 3-connected matroid in M has N as a proper minor. Checking if a matroid is a splitter is apotentially infinite task. But the Splitter Theorem [6, Theorem 12.1.2] establishes that if every3-connected single-element extension and coextension of N does not belong to M , then N is asplitter for M .Let M be a matroid and X be a subset of the ground set E . The connectivity function λ isdefined as λ ( X ) = r ( X ) + r ( E − X ) − r ( M ). Observe that λ ( X ) = λ ( E − X ). For k ≥
1, apartition (
A, B ) of E is called a k -separation if λ ( A ) ≤ k − | A | , | B | ≥ k . If λ ( A ) = k −
1, wecall (
A, B ) an exact k-separation . If λ ( A ) = k − | A | = k or | B | = k , we call ( A, B ) a minimalexact k-separation . For n ≥
2, we say M is n-connected if M has no k -separation for k ≤ n −
1. A k -connected matroid is internally ( k + 1) -connected if it has no non-minimal exact k -separations.Let M be a matroid in M with an N -minor and let N have an exact k -separation ( A, B ). Ifthere exists a k -separation ( X, Y ) of M such that A ⊆ X and B ⊆ Y , then we say the k -separation( A, B ) of N is induced in M . Suppose M is a k -connected matroid with a k -connected minor N such that N has a non-minimal exact k -separation ( A, B ). We call N a k -decomposer for M having ( A, B ) as an inducer, if M has a non-minimal exact k -separation ( X, Y ) such that A ⊆ X and B ⊆ Y . If ( A, B ) is not induced in M , then we say M bridges the k -separation ( A, B ) in N .Define k M ( A, B ) = min { λ M ( X ) | A ⊆ X ⊆ E ( M ) − B } . Thus M bridges ( A, B ) if and only if k M ( A, B ) ≥ k .The next theorem appears in [3, Theorem 1.2]. It gives a sufficient conditions to determinewhen an exact k -separation ( A, B ) in N is induced in M . Note that no isomorphism is involvedin the calculations of extensions and coextensions. Theorem 2.1.
Let N be a simple and cosimple matroid in M with an exact k -separation ( A, B ) ,such that A is the union of circuits and the union of cocircuits. Suppose M ∈ M . (i) If M is a simple single-element extension of N such that M \ e = N (no isomorphism isinvolved), then λ M ( A ) = k − or λ M ( A ∪ e ) = k − . (ii) If M is a cosimple single-element coextension of N such that M/f = N , then λ M ( A ) = k − or λ M ( A ∪ f ) = k − . (iii) If M is a cosimple single-element coextension of a Type (i) matroid or a simple single-element extension of a Type (ii) matroid, then M satisfies one of the following conditions: (a) λ M/f ( A ) = k − and λ M \ e ( A ) = k − ; (b) If λ M/f ( A ) = k − and λ M \ e ( A ∪ f ) = k − , then λ M ( A ∪ f ) = k − or { e, f, g } isa triad or triangle with g ∈ A ; PLITTERS AND DECOMPOSERS FOR BINARY MATROIDS 5 (c) If λ M/f ( A ∪ e ) = k − and λ M \ e ( A ) = k − , then λ M ( A ∪ e ) = k − or { e, f, g } is atriad or triangle with g ∈ A ; or (d) If λ M/f ( A ∪ e ) = k − and λ M \ e ( A ∪ f ) = k − , then { e, f, g } is a triangle or triadin M with g ∈ A .Then the k -separation ( A, B ) of N is induced in M for every M ∈ M with N as a minor. Practically speaking we are interested in 3-decomposers for classes of GF ( q )-representable ma-troids closed under minors and isomorphism because we have a “Splitter Theorem” only for 3-connected matroids. A simple matroid is 3-connected if λ ( A ) ≥ A, B ) with | A | ≥ | B | ≥
3. If N is 3-connected, a simple single-element extension and a cosimplesingle-element coextension are also 3-connected. A 3-connected matroid is internally -connected if λ ( A ) ≥ A, B ) with | A | ≥ | B | ≥
4. In this case λ ( A ) = 2 is allowedonly when either | A | or | B | has size at most 3. Suppose M is a 3-connected matroid having a3-connected minor N and N has a non-minimal exact 3-separation ( A, B ). If N is a for M , then M has a 3-separation ( X, Y ) such that A ⊆ X and B ⊆ Y . In this case λ ( X ) = 2and | X | ≥ | Y | ≥
4. Hence, if N is a 3-decomposer for M , then M is not internally 4-connected.The converse is not true.Next, we discuss the situation when N has two distinct non-minimal exact 3-separations ( A , B )and ( A , B ) where both A and A are unions of circuits and unions of cocircuits and both ( A i , B i )satisfy Theorem 2.1(i, ii). If both satisfy Theorem 2.1(i, ii) in such a way that λ ( A i ) = 2, then aone-element check suffices and we can conclude that both ( A , B ) and ( A , B ) are induced in allmatroids in M with an N -minor. Problems arise when λ ( A i ∪ e ) = 2. Now, suppose further that( A i , B i ) do not satisfy Condition (iii) for some of the coextension rows that can be added to Type(i) matroids. Then we cannot conclude that ( A i , B i ) are induced in all matroids in M with an N -minor. Note that, if M is not a bridging matroid, Condition (iii) fails because λ ( A ∪ e } ) = 2or λ ( A ∪ f } ) = 2, as the case may be. It is still true that λ ( A ∪ { e, f } ) = 2 (or else it would bea bridging matroid). Let B i be the set of “bad” coextension rows that cause Condition (iii) tofail for ( A i , B i ), where i = { , } . As long as B and B are disjoint, we can conclude that either( A , B ) or ( A , B ) is induced in M without being able to specify which one exactly.The next corollary follows from Theorem 2.1 and the above discussion. Further we hypothesizethat N is self-dual so the argument for the simple single-element extension for Type (ii) matroidsfollows by duality. Corollary 2.2.
Let N be a self-dual simple and cosimple matroid in M with two non-minimalexact -separations ( A , B ) and ( A , B ) , such that each of A and A are the union of circuitsand the union of cocircuits. Suppose M ∈ M is a Type (i) or (ii) matroid and condition (i) and(ii) hold for both ( A , B ) and ( A , B ) in such a way that if λ ( A i ∪ x ) = 2 , then λ ( A j ) = 2 for i, j ∈ { , } and x ∈ { e, f } . If M is a cosimple single-element coextension of a Type (i) matroidsuch that condition (iii) is satisfied by either ( A , B ) or ( A , B ) , then either ( A , B ) or ( A , B ) is induced in M for every M ∈ M with N as a minor. (cid:3) We end this section by describing our method for calculating extensions and coextensions. Let N be a GF ( q )-representable n -element rank- r matroid represented by the matrix A = [ I r | D ] over GF ( q ). The columns of A may be viewed as a subset of the columns of the matrix that representsthe projective geometry P G ( r − , q ). Let M be a simple single-element extension of N over GF ( q ). Then N = M \ e and M may be represented by [ I r | D ′ ], where D ′ is the same as D , butwith one additional column corresponding to the element e . The new column is distinct fromthe existing columns and has at least two non-zero elements. If the existing columns are labeled { , . . . , r, . . . , n } , then the new column is labeled ( n + 1). SPLITTERS AND DECOMPOSERS FOR BINARY MATROIDS
Suppose M is a cosimple single-element coextension of N over GF ( q ). Then N = M/f and M may be represented by the matrix [ I r +1 | D ′′ ], where D ′′ is the same as D , but with one additionalrow. The new row is distinct from the existing rows and has at least two non-zero elements. Thecolumns of [ I r +1 | D ′′ ] are labeled { , . . . , r + 1 , r + 2 , . . . , n, n + 1 } . The coextension element f corresponds to column r + 1. The coextension row is selected from P G ( n − r, q ), which meansthere could be a much larger selection of row vectors for the coextension. We can visualize the newelement f as appearing in the new dimension and lifting several points into the higher dimension.Observe that f forms a cocircuit with the elements corresponding to the non-zero elements in thenew row. Note that in [ I r +1 | D ′′ ] the labels of columns beyond r are increased by 1 to accomodatethe new column r + 1.We refer to the simple single-element extensions of N as Type (i) matroids and the cosimplesingle-element coextensions of N as Type (ii) matroids. The structure of type (i) and Type (ii)matroids are shown in Figure 1. Figure 1.
Structure of Type (i) and Type (ii) matroidsOnce the simple single-element extensions (Type (i) matroids) and cosimple single-elementcoextensions (Type (ii) matroids) are determined, the number of permissable rows and columnsgive a bound on the choices for the cosimple single-element extensions of the Type (i) matroidsand the simple single-element extensions of the Type (ii) matroids, respectively.The structure of the cosimple single-element coextensions of a Type (i) matroid and the simplesingle-element extensions of a Type (ii) matroid are shown in Figure 2.
Figure 2.
Structure of M , where | E ( M ) − E ( N ) | = 2When computing the cosimple single-element coextension of a Type (i) matroid, there are threetypes of rows that may be inserted into the last row.(i) rows that can be added to N to obtain a coextension with a 0 or 1 as the last entry (or asmany as the entries in GF ( q ) for higher order fields);(ii) the identity rows with a 1 in the last position; and PLITTERS AND DECOMPOSERS FOR BINARY MATROIDS 7 (iii) rows “in-series” to the right-hand side of the matrix with the last entry reversed.When computing the simple single-element extension of a Type (ii) matroid, there are three typesof rows that may be inserted into the last column.(i) columns that can be added to N to obtain an extension with a 0 or 1 as the last entry (oras many as the entries in GF ( q ) for higher order fields);(ii) the identity columns with a 1 in the last position; and(iii) columns “in-parallel” to the right-hand side of matrix with the last entry reversed.3. Proof of the main theorem
Proof of Theorem 1.1.
Let M be a 3-connected non-regular matroid in EX [ S , S ∗ ]. Then M has an F ∗ -minor. To get the 3-connected single-element extensions of F ∗ , compare the columnsin F ∗ to the columns in P G (3 ,
2) shown below.
P G (3 ,
2) = I Add missing columns one by one and group them into isomorphism classes. This gives us the twonon-isomorphic single-element extensions AG (3 ,
2) and S shown below. Only one column from P G (3 ,
2) (namely [1110]) may be added to F ∗ to obtain AG (3 , S [6, 12.2.4]. Claim 1. S is a -decomposer for EX [ P , P ∗ ]. Proof. S has a non-minimal exact 3-separation ( A, B ) where A = { , , , } and λ ( A ) = 2.Observe that, S has only two non-isomorphic single-element extensions P and Z . Moreover,only one column [1110] may be added to obtain Z and the remaining columns from P G (3 ,
2) give P . It is easy to check that λ ( { , , , } ) = 2 for this matrix representing Z . Since S is self-dual, Z ∗ is the only single-element coextension in EX [ P , P ∗ ] and it is obtained by adding only onerow [1110]. We can check that λ ( { , , , } ) = 2 in this coextension. By Theorem 2.1, S is a3-decomposer for EX [ P , P ∗ ]. Claim 2. P or P ∗ are -decomposers for EX [ S , S ∗ , E , E ] . Moreover, E is internally -connected. Proof.
Observe that P has three non-minimal exact 3-separations. They are ( A, B ), ( A , B ),and ( A , B ), where A = { , , , } , A = { , , , } and A = { , , , } . While A is a circuitand a cocircuit B , B , and B are not unions of cocircuits and A and A are not unions ofcircuits. Thus the only candidate for a non-minimal exact 3-separation is ( A, B ). Consider thesingle-element extensions of P : • adding column [1110] gives D ; • adding any one of columns [1001], [0101], [0110], or [1010] gives S ; and • adding column [0011] gives D .For the first and third extension, λ ( { , , , } ) = 2. P has 8 non-isomorphic cosimple single-element coextensions. They are obtained by adding the following columns: • [11000] or [11111] gives E ; • [11011] or [11100] gives E ; • [11001] or [11101] gives E ; • [01001], [01010], [01101], [01110], [10001], [10010], [10101], or [10110] gives E ; SPLITTERS AND DECOMPOSERS FOR BINARY MATROIDS • [01011], [01100], [10011], or [10100] gives E ; • [00101] or [00110] gives E ; • [00111] gives E ∗ ; and • [00011] gives E .By checking each row, regardless of isomorphism, we see that λ ( { , , , } ) = 2 for all the rowsthat give E , E , E , E , E ∗ , and E . The result follows from Theorem 2.1. Further note that E and E are self-dual and E is internally 4-connected. Claim 3. E is a splitter for EX [ S , S ∗ ] . Proof.
It is straightforward to check that every single-element extension of E has a minorisomorphic to S . The single-element extensions are obtained by adding the following columns: • [00011], [00101], [10010], [10100] gives ( E , ext • [00110] or [10001] gives ( E , ext • [00111], [10011], [10101] or [10110] gives ( E , ext • [01001], [01100], [01111] or [11101] gives ( E , ext • [01010], [11000], [11011] or [11110] gives ( E , ext • [01011] or [11100] gives ( E , ext • [01101] gives ( E , ext E is self-dual, every single-element coextension has a minor isomorphic to S ∗ . It followsthat E is a splitter for EX [ S , S ∗ ].Returning to the proof of Theorem 1.1 consider the simple single-element extensions and cosim-ple single-element coextensions of E . Observe that • adding column [00110] [10110] gives A ; • adding column [01111] [11100] gives B ; • adding column [11000] gives C ; and • adding column [11011] gives T /e .Adding any other column to E gives a matroid with an S -minor. Similarly, • adding row [00110] or [10001] gives A ∗ ; • adding row [11001] [11100] gives B ∗ ; • adding row [11000] gives C ∗ ; and • adding row [01010] gives T \ e Adding any other row to E gives a matroid with an S -minor: From T /e we get T whichis a splitter for EX [ S , S ∗ ]. See [1] for the details. Thus if M has a T /e as a minor, then M ∼ = T /e, T \ e, T .Next, observe that E has two non-minimal exact 3-separations ( A , B ) and ( A , B ), where A = { , , , , , } and A = { , , , , , } . Moreover, A and A are both union of circuitsand union of cocircuits. Note that B and B do not meet this condition. A = { , , } ∪ { , , , } and A = { , , } ∪ { , , , } A = { , , } ∪ { , , , } and A = { , , } ∪ { , , , } Using Theorem 2.1 and Corollary 2.2 we will prove that every matroid in EX [ S , S ∗ ] with an E -minor has a non-minimal exact 3-separation induced by either ( A , B ) or ( A , B ). As wesee in the proof, we cannot determine which of the two 3-separations is induced in a particularmatroid, just that at least one of them is induced. Claim 4.
Suppose M ∈ EX [ S , S ∗ ] has an E -minor. Then M has a non-minimal exact -separation induced by either ( A , B ) or ( A , B ) . PLITTERS AND DECOMPOSERS FOR BINARY MATROIDS 9
Proof.
Table 1a and 1b show that Theorem 2.1(i, ii) are satisfied.
Name Extension Columns A = { , , , , , } A = { , , , , , } A α = [00110] λ { , , , , , } = 2 λ { , , , , , , } = 2 β = [10110] λ { , , , , , , } = 2 λ { , , , , , } = 2 A γ = [01111] λ { , , , , , , } = 2 λ { , , , , , } = 2 δ = [11100] λ { , , , , , } = 2 λ { , , , , , , } = 2 A ǫ = [11000] λ { , , , , , } = 2 λ { , , , , , } = 2 Table 1a: Simple single-element extensions of E in EX [ S , S ∗ ] Name Coextension Rows A = { , , , , , } A = { , , , , , } A ∗ a = [00110] λ { , , , , , } = 2 λ { , , , , , , } = 2 b = [10001] λ { , , , , , , } λ { , , , , , } = 2 A ∗ c = [11001] λ { , , , , , , } = 2 λ { , , , , , } = 2 d = [11100] λ { , , , , , } = 2 λ { , , , , , , } = 2 A ∗ e = [11000] λ { , , , , , } = 2 λ { , , , , , } = 2 Table 1b: Cosimple single-element coextensions of E in EX [ S , S ∗ ]Next, we compute the cosimple single-element coextensions of the Type (i) matroids A , B , and C . As explained in Section 2, there are three types of rows that can be added: rows a to e witha 0 or a one in the last entry; the identity rows with a 1 in the last entry; and rows in-serieswith rows in the matrix with the last entry switched. We handle ( A , B ) and ( A , B ) separately.However, notice that A with column [10110] and B with column [01111] have coextensions thatexhibit similar behavior. Coextension Rows S , S ∗ A = { , , , , , } A with column [10110] a = [001100] No λ { , , , , , , } = 2and a ′ = [001101] Yes - B with column [01111] b = [100010] No Bad row b ′ = [100011] No Bad row c = [110010] No Bad row c ′ = [110011] No Bad row d = [111000] Yes - d ′ = [111001] No λ { , , , , , , } = 2 e = [110000] No Bad row e ′ = [110001] No λ { , , , , , , } = 23 ′ = [110100] Yes -4 ′ = [111100] Yes -9 ′ = [001001] Yes -10 ′ = [000101] Yes - A with column [00110] b = [100010] No λ { , , , , , , } = 2and b ′ = [100011] Yes - B with column [11100] c = [110010] No λ { , , , , , , } = 2 c ′ = [110011] yes - C with column [11000] b = [100010] No λ { , , , , , , } = 2 b ′ = [100011] No Bad row c = [110010] No λ { , , , , , , } = 2 c ′ = [110011] No Bad rowTable 2a: Cosimple single-element coextensions of A , A , A in EX [ S , S ∗ ] with respect to ( A , B ) Coextension Rows S , S ∗ A = { , , , , , } A with column [00110] a = [001100] No Bad rowand a ′ = [001101] No Bad row B with column [11100] b = [100010] No { , , , , , , } b ′ = [100011] Yes - c = [110010] No { , , , , , , } c ′ = [110011] yes - d = [111000] No Bad row d ′ = [111001] No Bad row e = [110000] No { , , , , , , } e ′ = [110001] No Bad row3 ′ = [110100] Yes -4 ′ = [111100] Yes -9 ′ = [001001] Yes -10 ′ = [000101] Yes - A with column [10110] a = [001100] No { , , , , , , } and b ′ = [100011] Yes - B with column [01111] c = [110010] Yes - c ′ = [110011] No { , , , , , , } C with column [11000] a = [001100] No λ { , , , , , , } = 2 a ′ = [001101] No Bad row d = [111000] No λ { , , , , , , } = 2 d ′ = [111001] No Bad rowTable 2b: Cosimple single-element coextensions of A , A , A in EX [ S , S ∗ ] with respect to ( A , B ) Let us call the rows that give matrices in EX [ S , S ∗ ] satisfying conditions in Theorem 2.1(iii)for a particular 3-separation ( A i , B i ) good rows for ( A i , B i ) and those that don’t, but are notbridging matroids, bad rows for ( A i , B i ). Observe from Table 2a and 2b for 3-separation ( A , B ), A with column [001100] and B with column [01111] have the property that all rows to be addedare good. Similarly for 3-separation ( A , B ), A with column [10110] and B with column [01111]have the property that all rows to be added are good. Moreover, for C with column [11000] thesets of bad rows for ( A i , B i ) are disjoint. It follows from Corollary 2.2 that either ( A , B ) or( A , B ) is induced in all matroids in EX [ S , S ∗ ] with an E -minor.Lastly, although we do not need to check Theorem 2.1(iv) since E is self-dual, we did verify itby repeating all the calculations for the simple single-element extensions of A ∗ , B ∗ , and C ∗ . Thiscompletes the proof of Theorem 1.1. (cid:3) References (1) Kingan, S. R. (1997) A generalization of a graph result by D. W. Hall,
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