A Decomposition Theorem for Binary Matroids with no Prism Minor

Transcription

1 DOI /s ORIGINAL PAPER A Decomposition Theorem for Binary Matroids with no Prism Minor S. R. Kingan Manoel Lemos Received: 21 March 2012 / Revised: 13 January 2013 Springer Japan 2013 Abstract The prism graph is the dual of the complete graph on five vertices with an edge deleted, K 5 \e. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac s infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of F 7 with itself across a triangle with an element of the triangle deleted; it s rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3- separation in P 9. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, F 7 and PG(3, 2), respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of R 10, the unique splitter for regular matroids. As a corollary, we obtain Mayhew and Royle s result identifying the binary internally 4-connected matroids with no prism minor Mayhew and Royle (Siam J Discrete Math 26: , 2012). The first author is partially supported by PSC-CUNY grant number The second author is partially supported by CNPq under Grant number / S. R. Kingan (B) Department of Mathematics Brooklyn College, City University of NewYork, Brooklyn, NY 11210, USA skingan@brooklyn.cuny.edu M. Lemos Departamento de Matematica, Universidade Federal de Pernambuco, Recife Pernambuco , Brazil manoel@dmat.ufpe.br

2 Keywords Binary matroids Decomposition of matroids Excluded-minors Prism graph 1 Introduction In a decomposition result, a more complicated matroid is broken down into simpler components. The fact that such simplifications exist is surprising and indicative of deep order in the structure of infinite classes of matroids. In 1980 Seymour decomposed the class of regular matroids, begining a flourishing genre of structural results [9]. A matroid is regular if it has no minor isomorphic to the Fano matroid F 7 or its dual F 7. To decompose regular matroids, he developed the Splitter Theorem, a Decomposition Theorem, and the notion of 3-sums. The Splitter Theorem describes how 3-connected matroids can be systematically built-up and the Decomposition Theorem describes the conditions under which a specific type of separation in a matroid gets carried forward to all matroids containing it. The proof of the decomposition of regular matroids consists of three main parts. The first part establishes that a 3-connected regular matroid is graphic or cographic or has a minor isomorphic to R 10 or R 12. The matroid R 10 is a splitter for regular matroids. This means no 3-connected regular matroid contains it (other than R 10 itself). So the building-up process stops at R 10. The second part establishes that R 12 has a non-minimal exact 3-separation that carries forward in all 3-connected regular matroids containing it. The third part establishes that 3-connected regular matroids with an R 12 -minor can be pieced together from graphic and co-graphic matroids using the operation of 3-sums. It suffices to focus on the 3-connected members of a class because matroids that are not 3-connected can be pieced together from 3- connected matroids using the operations of 1-sum and 2-sum. In this paper we present the decomposition of binary matroids with no minor isomorphic to the prism graph. The prism graph, denoted as (K 5 \e), is shown in Fig. 1. To decompose this class we used the Splitter Theorem [9] and a decomposition theorem by Mayhew, Royle, and Whittle [4]. The class of binary matroids with no prism minor is quite different from the class of regular matroids, but also similar in the sense that there are several special matroids in it and one of them has a separation that carries forward. The role of R 12 is played by the non-regular matroid P 9. The matroid terminology follows Oxley [7]. We should note that the matroid corresponding to the matrix labeled A is called M[A] and not just A. However, we refer to large numbers of matrices in this paper and with the reader s understanding treat the matrix and matroid as synonymous. Let M be a matroid and X be a subset of the ground set E. Theconnectivity function λ is defined as λ(x) = r(x) + r(e X) r(m). Observe that λ(x) = λ(e X). Fork 1, a partition (A, B) of E is called a Fig. 1 The prism graph and its matrix representation

3 k-separation if A k, B k, and λ(a) k 1. When λ(a) = k 1, we call (A, B) an exact k-separation. When λ(a) = k 1 and A =k or B =k we call (A, B) a minimal exact k-separation. Forn 2, we say M is n-connected if M has no k-separation for k n 1. A matroid is internally n + 1-connected if it is n-connected and has no non-minimal exact n-separations. In particular, a simple matroid is 3-connected if λ(a) 2 for all partitions (A, B) with A 3 and B 3. A 3-connected matroid is internally 4-connected ifλ(a) 3 for all partitions ( A, B) with A 4and B 4. Let M be a class of matroids closed under minors and isomorphisms. Let k 1 and N be a matroid belonging to M having an exact k- separation (A, B).LetM M having an N-minor. We say that N is a k-decomposer for M having (A, B) as an inducer provided M has a k-separation (X, Y ) such that A X and B Y. For r 3, let W r denote the wheel with r spokes, and for p 3, let K 3,p denote the complete bipartite graph with three vertices in one class and p vertices in the other class. Let K 3,p, K 3,p, and K 3,p denote the graphs obtained from K 3,p by adding one, two, and three edges, respectively, joining vertices in the class containing three vertices. Let Z r denote the (2r + 1)-element rank-r non-regular matroid represented by the binary matrix [I r D] where D has r +1 columns labeled b 1,...,b r, c r. The first r columns in D have zeros along the diagonal and ones elsewhere. The last column is all ones. None of these infinite families have a prism minor. Some matroids like R 10, the splitter for regular matroids, play a central role in structural results. In addition to R 10, this class contains four such significant matroids, P 9, E 5, D 1 and R 17. Matrix representations are shown below. R 10 = I P 9 = I D 1 = I E 5 = I R = I The matroid P 9 is a binary non-regular 9-element rank-4 matroid. It is the generalized parallel connection of F 7 and W 3 across a triangle, with an element of the triangle deleted (denoted as P (F 7, W 3 )\z). It has a non-minimal exact 3-separation (A, B), where A ={1, 2, 5, 6}. The matroid P 9 plays the same role in this paper as R 12 does for regular matroids. The matroid D 1 is a single-element extension of P 9.Itisisomorphic to P (F 7, F 7 )\z. The matroid E 5 is a binary non-regular 10-element rank-5 matroid that is self-dual and internally 4-connected. It is a single-element extension of P 9 and M(K 3,3). The matroid R 17 is a 17-element rank 5 matroid that is an extension of both E 5 and R 10 [2]. The next result is the main theorem of this paper.

4 Theorem 1.1 Suppose M is a 3-connected binary matroid with no M (K 5 \e)-minor. Then either P 9 is a 3-decomposer for M or M is isomorphic to one of the following matroids: (i) M(W r ) for some r 3, M(K 3,p ),M(K 3,p ),M(K 3,p ) or M(K 3,p ),forsome p 3; or (ii) F 7,F7,Z r,zr,z r \b r,orz r \c r,forsomer 4; (iii) D1 ;or (iv) PG(3, 2) or R 17 or one of their 3-connected restrictions. As a technicality, it may look to the reader like some obvious matroids are missing in the above list. However, note that Z 4 \c 4 = AG(3, 2), Z4 \b 4 = S8, M(K 5 ), M(K 5 \e), M (K 3,3 ), F 7, P 9 and D 1 are restrictions (deletion-minors) of PG(3, 2). The matroids R 10, P 9, and E 5 are restrictions of R 17. So we do not have to list these matroids explicitly. The next result by Mayhew and Royle in [5] is a corollary of Theorem 1.1. The matroid they call AG(3, 2) U 1,1 is R 17. It should be noted that Mayhew and Royle use a completely different method as compared to Theorem 1.1. Corollary 1.2 M is an internally 4-connected binary matroid with no M (K 5 \e)- minor if and only if M is isomorphic to an internally 4-connected restriction of F 7, PG(3, 2) or R 17. To make the paper complete, we determine the class of binary matroids with no M(K 5 \e)-minor and the class with neither M(K 5 \e) nor M (K 5 \e)-minor. Corollary 1.3 Suppose M is a 3-connected binary matroid with no M(K 5 \e)-minor. Then either P 9 or P9 is a 3-decomposer or M is isomorphic to one of the following matroids: (i) M (K 3,3 ) or M(W r ),forr 3; (ii) F 7,F7,Z r,zr,z r \b r,orz r \c r,forr 4; or (iii) PG(3, 2) or R17 or one of its 3-connected contraction-minors. Corollary 1.4 Suppose M is a 3-connected binary matroid with neither M(K 5 \e)- nor M (K 5 \e)-minor, then either P 9 is a 3-decomposer for M or M is isomorphic to M(K 3,3 ),M (K 3,3 ),M(W r ) for r 3, Z r,zr,z r \b r,z r \c r,forr 4, F 7,F7,P 9, P9,D 1,D1,R 10, ore 5. In Sect. 2 we give some preliminaries and in Sect. 3 we prove the Decomposition Lemma that forms a key component of the proof of Theorem 1.1. Finally, in Sect. 4 we prove Theorem 1.1 and the corollaries. 2 Preliminaries The origin of this excluded minor problem can be traced to 1963 when Dirac determined the extremal graphs without two vertex disjoint cycles [1]. Excluding two vertex-disjoint cycles in a 3-connected graph is equivalent to excluding (K 5 \e) as a minor. So, essentially Dirac determined the 3-connected graphs with no prism minor.

5 Theorem 2.1 Asimple3-connected graph has no minor isomorphic to (K 5 \e) if and only if it is isomorphic to W r for some r 3, K 5,K 5 \e, K 3,p,K 3,p,K 3,p or K 3,p, for some p 3. In 1984 Robertson and Seymour published a note where they proved that a simple 3-connected graph with at least four vertices has no minor isomorphic to K 5 \e if and only if it is isomorphic to (K 5 \e), K 3,3,orW r for some r 3[8]. In 1996 Kingan characterized the 3-connected regular matroids with no minor isomorphic to M (K 5 \e) [2,2.1]. Theorem 2.2 A 3-connected regular matroid has no minor isomorphic to M (K 5 \e) if and only if it is isomorphic to M(W r ) for some r 3, M(K 5 ),M(K 5 \e), M (K 3,3 ), M(K 3,p ),M(K 3,p ),M(K 3,p ),orm(k 3,p ),forsomep 3,orR 10. Thus in order to characterize the class of binary matroids with no prism minor we may focus only on non-regular matroids. Tutte proved that a binary matroid is nonregular if and only if it has no minor isomorphic to F 7 or F7. This makes F 7 or F7 the starting point of our investigation. Observe that F 7 = PG(2, 2) and as such has no extensions in the class of binary matroids. Coextensions of F 7 are duals of extensions of F7. Thus we may focus on the extensions of F 7. Observe that AG(3, 2) and S 8 are the two non-isomorphic 3-connected single-element extensions of F7. Since they are self-dual, they are also the coextensions of F 7. The matroid S 8 has two non-isomorphic 3-connected single-element extensions P 9 and Z 4 and AG(3, 2) has one 3-connected single-element extension, Z 4. As noted earlier, P 9 has a non-minimal exact 3-separation (and consequently so does P9 ). The matroid P 9 first appeared in [6] where Oxley characterized the 3-connected binary non-regular matroid with no minors isomorphic to P 9 or P9. Theorem 2.3 A 3-connected binary non-regular matroid has no minor isomorphic to P 9 or P9 if and only if it is isomorphic to F 7,F7,Z r,zr,z r \b r,orz r \c r,forsome r 4. It is easy to show that Z r and Zr do not have a prism minor. To prove Theorem 2.3, Oxley proves that for r 4, Z r, Zr, Z r \c r, and Z r \b r have no M(W 4 )-minor [6, Theorem 2.1]. Since M (K 5 \e) and M(K 5 \e)-minor have an M(W 4 )-minor, we may conclude that Z r, Zr, Z r \c r, and Z r \b r have no minor isomorphic to M (K 5 \e) nor M(K 5 \e). As a consequence we may conclude a binary non-regular matroid with no prism minor is either one of the infinite families mentioned in Theorem 2.3 or it has a P 9 -orp9 -minor. Then, we will prove the stronger statement that P 9 is not relevant and, in fact, P 9 is the required 3-decomposer (with one exception). Like R 12, P 9 has a non-minimal exact 3-separation in it. However, unlike R 12, the separation in P 9 does not extend to all the 3-connected binary matroids with no prism minor containing it. Nonetheless, we are able to identify the exceptions. Since the prism graph has rank 5, all the binary non-regular 3-connected rank 4 matroids have no prism minor. So PG(3, 2) and all of its deletion minors have no prism minor. We will prove that besides one 10-element rank-6 matroid (D1 ), all the exceptions have rank at most 5. This result (Decomposition Lemma) forms a key component of the proof of the main

6 theorem. Subsequently, a detailed analysis of rank 5 binary matroids with no prism minor revealed that all of them are restriction minors of the 17-element matroid R 17. We need two structural results to prove the Decomposition Lemma. We prove that P 9 or P9 are 3-decomposers for a certain class of matroids using the following result by Mayhew, Royle, and Whittle in [4, 2.10]. Theorem 2.4 Suppose M is a class of matroids closed under minors and isomorphism and let N M be a 3-connected matroid with E(N) 8 and a non-minimal exact 3-separation (A, B) where A is a 4-element circuit and a cocircuit. If A is a circuit and cocircuit in every 3-connected single-lement extension and coextension of N in M, then N is a 3-decomposer for every matroid in M with an N-minor. The significance of the above result is that it makes it easy to determine whether or not a non-minimal exact 3-separation carries forward. To see this compare this criteria to the original criteria in Seymour s Decomposition Theorem [8, 9.1]. We also need the Splitter Theorem. The following result is a version of the Splitter Theorem given in [7, (12.2.1)]. It may be worth noting that the Strong Splitter Theorem [3] was discovered while proving the main theorem of this paper. Theorem 2.5 (Splitter Theorem) Suppose N is a 3-connected proper minor of a 3- connected matroid M such that, if N is a wheel or whirl then M has no larger wheel or whirl-minor, respectively. Then, there is a sequence M 0,...,M n of 3-connected matroids with M 0 = N, Mn = M and for i {1,...,n}, M i is a single-element extension or coextension of M i 1. 3 The Decomposition Lemma In this section we state and prove the key component of the main theorem (Theorem 1.1). The argument is matrix theoretic and has a computational flavor. Tables of single-element extensions are in the Appendix. A computer is used for finding isomorphisms between single-element extensions. The interesting thing about isomorphismchecking is that once an isomorphism is found, no matter how it is found, whether by hand or software, checking it by hand is easy. As such the proof doesn t rely on a computer for verification, especially since the number of isomorphisms to be verified is quite small due to the theoretical arguments. Theorem 3.1 (Decomposition Lemma) Suppose M is a 3-connected binary nonregular matroid with no M (K 5 \e)-minor. Then one of the following holds: (i) M is isomorphic to F 7,F7,Z r,zr,z r \b r,orz r \c r,forsomer 4; (ii) P 9 is a 3-decomposer for M; (iii) M is isomorphic to D1 ;or (iv) M has rank at most 5. Proof Theorem 2.3 implies that if M has no P 9 nor P9 -minor, then M is isomorphic to F 7, F7, Z r, Zr, Z r \c r,orz r \b r for r 4. Thus we may assume that M has a P 9

7 or P 9 -minor. The matroid P 9 has three simple non-isomorphic binary single-element extensions. Using the matrix representation for P 9 given in Section 1, adding column [1110] gives D 1, adding any one of columns [1001], [0101], [0110], or[1010] gives D 2, and adding column [0011] gives D 3. This is concisely summarized in Table 1 and representative matrices for D 1, D 2, and D 3 are given below. Further, note that Table 1 gives all the extensions of P 9. Columns in bold are the ones used to form the matrices. The final three rank 4 matrices are PG(3, 2), PG(3, 2)\e and PG(3, 2)\{e, f } D 1 = I D 2 = I D 3 = I P 9 has eight cosimple non-isomorphic single-element coextensions (see Table 2). When coextending a rank-4 matrix the column [0, 0, 0, 0, 1] is added as the fifth element and a new row is added at the bottom of the right hand side of the matrix. The coextended element is column E 1 = I E = I E = I E 4 = I E = I E = I E6 = I E = I As mentioned earlier, P 9 has a non-minimal exact 3-separation (A, B) where A = {1, 2, 5, 6} is both a circuit and a cocircuit. It is easy to check that the set A = {1, 2, 5, 6} is both a circuit and a cocircuit in D 1 and D 3, whereas D 2 is internally 4- connected. The set A ={1, 2, 5, 6} corresponds to A ={1, 2, 6, 7} in the coextension since the fifth column is the coextended element. It can be checked that {1, 2, 6, 7} is both a circuit and a cocircuit in E 1, E 2, E 3, E 6, E6, and E 7. Further note that E 4 and E 5 are self-dual. Theorem 2.4 implies that if M has a P 9 -minor, but no D 2, D2, E 4, or E 5 -minor, then P 9 or P9 is a 3-decomposer for M. Observe from Tables 3 and 4 that D 2 has an M(K 5 \e)-minor (and therefore D2 has an M (K 5 \e)-minor) and E 4 has an M (K 5 \e)-minor. Therefore we may assume going forward that M has a D 2 or E 5 -minor.

8 Next, we will prove that if M has a P9 -minor, but no E 5-minor, then either P 9 is a 3-decomposer for M or M = D1. Suppose M is an extension of P 9. The extensions of P9 are the duals of the extensions of P 9. From Table 2 they are E 1, E 2, E 3, E 4, E 5, E 6, E6 and E 7. All of these matroids, except E 7,haveaP 9-minor, since E 1, E 2, E 3, E 4, and E 5 are self-dual and E 6 and E6 are both coextensions of P 9. Moreover, since E 4 and E7 have an M (K 5 \e)-minor and E 5 is excluded by hypothesis, we may conclude that P 9 is a decomposer for M. Suppose M is a coextension of P9. Then since coextensions of P 9 are duals of extensions of P 9, M is isomorphic to D1, D 2, and D 3. Of these D 2 and D 3 have an M (K 5 \e)-minor. So M has a minor isomorphic to D1.IfM = D1, then we are done. From Table 1 we see that D 1 is formed by adding just one column to P 9, namely [1110], so any extension of D 1 will have a D 2 or D 3 -minor. Thus any coextension of D1 will have a D 2 or D 3 -minor. The extensions of D 1 are the duals of the coextensions of D 1. Observe from Table 5 that all except the second coextension have a P9 -minor. Since the second coextension has an E 7 -minor, its dual has an E7 -minor. Thus we may conclude that M = D1 or P 9 is a 3-decomposer for M. To complete the proof, we must show that if M has an E 5 or D 2 -minor, then r(m) 5. Claim 1 If M has an E 5 -minor and no M (K 5 \e)-minor, then r(m) 5. Proof of Claim 1. Suppose M is a rank 6 coextension of E 5. Observe from Table 6 that E 5 is self-dual and has seven non-isomorphic binary 3-connected single-element extensions, all of which have a minor isomorphic to D 2. Since D 2 has an M (K 5 \e)- minor and E 5 is self-dual, all the coextensions of E 5 have an M (K 5 \e)-minor. Further, all the coextensions of E 5 except A, B, C have an E 4 -minor (and therefore an M (K 5 \e)-minor). Matrix representations for A, B, and C are given below A = I B = I C = I We will prove that if M is a single-element coextension of A, B, orc, then M has an M (K 5 \e)-minor. Note that until now the proof relied on correctly reading and interpreting tables of single-element extensions. From now on further analysis is needed. A partial matrix representation for M is shown in Fig. 2. Fig. 2 Structure of a coextension of A, B, C

9 There are three types of rows that may be inserted into the last row on the right-hand side of the matrix in Fig. 2. (i) rows that can be added to E 5 to obtain a coextension with no M (K 5 \e)-minor, witha0or1asthelastentry; (ii) the identity rows witha1inthelastposition; and (iii) rows in-series to the right-hand side of matrices A, B, C with the last entry reversed. There are no Type I rows since 3-connected binary single-element coextensions of E 5 have an M (K 5 \e)-minor. Type II rows are [100001], [010001], [001001], [000101], and [000011]. Type III rows are specific to the matrices A, B, C. For matrix A they are [1], [101101], [110110], [111101], [110000]. For matrix B they are [0], [101101], [110111], [111100], and [110000].ForC they are [0], [101100], [110111], [111101], and ]. Thus we see that only ten rows may be added for each of A, B, C. Table 7 shows that most of these rows result in matroids that are isomorphic to matroids with an M (K 5 \e)-minor. Only two coextensions must be specifically checked for an M (K 5 \e)-minor: (C, coextn9) and (C, coextn10). Observe that, (C, coextn9)/12\1 = E 4, and (C, coextn10)/12\10 = E 4. Since E 4 has an M (K 5 \e)-minor, we may conclude these matroids have it too. Next, let us compute the single-element extensions of A, B, and C with no M (K 5 \e)-minor. Table 8 implies that the only columns that can be added to E 5 to obtain a matroid with no M (K 5 \e)-minor are [00101], [00110], [01011], [01100] [10011], [11001], [11101]. Adding these columns gives us four non-isomorphic singleelement extensions of A, B, and C. TheyareD, E, F, and G shown below (all the extensions of E 5 are shown in Table 8) D = I E = I F = I G = I Suppose M is a single-element coextension of D, E, F, or G. Then the structure of M is shown in Fig. 3. Recall that there are no Type I rows to be added. Adding a Type II or III row (with the exception of [ ]) causes M\12 to be 3-connected (and there are no such matroids). So the only coextension to check is the one formed by adding row [ ]. LetD, E, F, and G be the coextension of D, E, F, and G, respectively, obtained by coextending with row [ ]. Then in each case we can find an E 4 minor. In particular, D /1\{3, 11} = E 4, E /1\{7, 11} = E 4, F /1\{7, 11} = E 4, and G /1\{7, 11} = E 4. Thus we may conclude that if M is a single-element coextension of D, E, F,orG, then M has an M (K 5 \e)-minor.

10 Fig. 3 Structure of a coextension of D, E, F, G Finally, observe that if M is a rank-6 coextension of E 5 of size n 13, then for some e {11,...,n 1}, M\e is 3-connected, and therefore has an M (K 5 \e)-minor. Thus we have shown that rank-6 coextensions of E 5 have an M (K 5 \e)-minor. Claim 1 follows from the Splitter Theorem (Theorem 2.5). Claim 2 If M has a D 2 -minor and no M (K 5 \e)-minor, then r(m) 5. Proof of Claim 2. Suppose M is a rank 5 coextension of D 2. Observe from Table 5 that all the single-element extensions of D 2, except for A, B, C, and Z have an E 4 -minor. Further, observe that Z is an extension of R 10 and E 7 (see Table 5) and as a result, Z has an E 7 -minor (and therefore an M (K 5 \e)-minor). It gives us a slight computational advantage to view Z as an extension of R 10 and to show that 3-connected binary non-regular coextensions of R 10 have an M (K 5 \e)-minor. In the representation of R 17 given in the introduction, R 10 is isomorphic to the first ten columns and Z is isomorphic to the first eleven columns. Let us take that as a representation of Z Z = I R 10 has two non-isomorphic binary 3-connected single-element extensions, Z and B, where Z is obtained by adding any one of the columns [01011], [01101] [10101] [10110] [11010] or [11111] and B is obtained by adding any one of the remaining columns. Observe that adding all of the above six columns to Z gives us R 17 \e. Adding one additional column gives us R 17. Let M be a single-element coextension of Z. As before, there are three types of rows that may be added to M. There are no Type I rows since coextensions of R 10 are B and Z, both of which have a D 2 -minor (and consequently an M (K 5 \e)-minor). Type II rows are [100001], [010001], [001001], [000101] and [000011]and Type III rowsare [100111], [110010], [111001], [011100] and [001110]. Observe that, adding any of the above ten rows to Z gives the same matroid up to isomorphic (see Table 7). Without loss of generality let M be obtained from Z by adding row [000011]. Then, M/1\7 = E 4. Therefore, every coextension of Z has an M (K 5 \e)-minor.

11 It is easy to check that Z has three non-isomorphic single-element extensions, namely, D and F mentioned above, and Y shown below Y = I Repeating the argument in Claim 1, we may only consider the coextension of Y formed by adding row [ ].LetY be this coextension. Then, Y /1\{2, 7} = E 4. So Y has an M (K 5 \e)-minor. Thus, we may conclude that rank 6 matroids with an R 10 -minor also have an M (K 5 \e)-minor. By the Splitter Theorem all coextensions of R 10 have an M (K 5 \e)-minor. Next, observe from Table 1 that D 2 has two single-element extensions X 1 and X 3 shown below: X 1 = I X 3 = I There are three types of rows that may be added to X 1 and X 3. (i) the rows that can be added to D 2 to obtain a coextension with no M (K 5 \e)- minor with a 0 or 1 in the last entry. (These are the rows corresponding to A, B, C, Z in Table 6.) (ii) the identity rows witha1inthelastposition; (iii) and the rows in-series to the right-hand side of matrices X 1 and X 3 with the last entry reversed. If M is the coextension obtained by adding the first type of row, then M\12 is isomorphic to a coextension of D 2.SoM is isomorphic to A, B, C, orz and therefore, M is either D, E, F, G or Y. Type II rows are [ ], [ ], [ ], [ ], [ ], [ ]. Type III rows for X 1 are [1], [101101], [110110], [111101], [110000] and for X 3 are [0], [101101], [110111], [111100], and [110000]. ForC they are [0], [101100], [110111], [111101], and ]. In each case we were able to find an M (K 5 \e)-minor (see Table 9). From Table 1 we see that X 1 and X 3 have two non-isomorphic single-element extensions Y 1 and Y 2. Suppose M is a coextension of Y 1 or Y 2. Then M has rank 5 and 13 elements. If we add Type I rows, then M\13 is 3-connected, and if we add Type II or III rows, then M\12 is 3-connected, except when the row added is [ ]. So only two matroids must be specifically checked for an M (K 5 \e)-minor. They are Y 1 with row [ ] and Y 2 with row [ ]. In both cases the resulting matroid has an M (K 5 \e)-minor. Finally, observe that if M is a coextension of D 2 with size n 13, then for some e {11,...,n 1}, M\e is 3-connected and therefore has

12 an M (K 5 \e)-minor. Thus we have shown that rank 6 coextensions of D 2 have an M (K 5 \e)-minor. Claim 2 follows from the Splitter Theorem. From Claim 1 and Claim 2 we may conclude that if M has an E 5 or D 2 -minor, then the rank of M is at most 5. 4 Proof of Theorem 1.1 and corollaries In this section we prove the main theorem of this paper and its corollaries. Proof of Theorem 1.1. If M is regular, then the result follows from Theorem 2.2, so we may assume M is non-regular. Theorem 3.1 implies that it is sufficient to focus on matroids with rank at most 5. Since M (K 5 \e) has rank 5, F 7 and all the extensions of F7 up to PG(3, 2) are in the excluded minor class (see Table 1). To complete the proof we must show that R 17 is the extremal rank 5 binary matroid with no M (K 5 \e)-minor. This is done by showing that if M is a rank 5 matroid with an E 5 -ord 2 -minor then M = R 17 or its 3-connected restrictions (except P9 because it has only 9 elements). Table 6 implies that the only columns that can be added to E 5 to obtain a matroid with no M (K 5 \e)-minor are those that give A, B, C. That is, columns [00101], [00110], [01011], [01100] [10011], [11001], [11101]. It is straightforward to check that adding all of these columns gives us a matroid isomorphic to R 17 (see Table 8). From the proof of Claim 2 of Theorem 3.1, we see that besides A, B, and C, the matroid Z is the only coextension of D 2 with no M (K 5 \e)-minor. As noted earlier, Z is an extension of R 10 and is obtained by adding any one of the columns [01011], [01101][10101][10110][11010] or [11111] to R 10. The only other extension of R 10 is B. Adding all of the above six columns to Z gives us R 17 \e. Adding one additional column (corresponding to extension B) gives us R 17. One final matter must be checked. It may be possible for R 17 or one of its deletionminors to be an extension of the graph (K 5 \e) + edge or the cograph (K 3,3 ). We must rule out this possibility. To do so, first observe that M (K 5 \e) has only two regular extensions (K 5 \e) + edge and (K 3,3 ). Second, observe that E 5 has no minor isomorphic to the prism graph or its dual. Third, Table 5 lists all the 3-connected deletion-minors of A, B, C, making it clear that they have no M (K 5 \e)-minor. Lastly, Table 8 gives the single-element extensions of A, B and C using columns [00101], [00110], [01011], [01100] [10011] [11001] and [11101] (the other columns give an E 4 minor, which has a prism minor). These columns give four 12-element extensions, D, E, F, and G. These 12-element matroids have no graphic nor cographic singleelement deletions. Therefore, all their extensions will have no graphic nor cographic single-element deletions. We are justified in addding all these columns to E 5 to get R 17. This completes the proof of Theorem 1.1. The proof of Corollary 1.2 follows immediately. Moreover, using Table 8, we can identify the internally 4-connected restrictions of R 17 as all, except C, G and K. Among restrictions of PG(3, 2) all except K 5 \e, S 8, AG(3, 2), P 9, Z 4, D 1, D 3, and X 2 are internally 4-connected. It should be noted that the main theorem in Mayhew and Royle has two parts [4, Theorem 1.1]. The first part is Corollary 1.2. The second part

13 states that if M is 3-connected and not internally 4-connected and M has an internally 4-connected minor with at least 6 elements not isomorphic to W 3, F 7, F 7,orK 3,3, then M is isomorphic to one of five sporadic matroids. The five sporadic matroids they refer to are C, G and K, an 11-element rank-5 single-element coextension and a 12-element rank-6 double-element coextension of K 5 [5, Table 3]. We end this paper by using the Decomposition Lemma to prove Corollary 1.3.The proof of Corollary 1.4 is an immediate consequence of Theorem 1.1 and Corollary 1.3. Note that in the case of Corollary 1.3, we can only conclude that either P 9 or P 9 are 3-decomposers, instead of the stronger statment that P 9 is a 3-decomposer. This is because E 7 has no M(K 5\e)-minor nor P 9 -minor. However, it does admit the 3-separation of its minor P 9. Proof of Corollary 1.3. Suppose M is a 3-connected binary non-regular matroid with no M(K 5 \e)-minor. If M is regular, then M is isomorphic to M (K 5 \e), M(K 3,3 ), M (K 3,3 ), M(W r ) for some r 3, or R 10. Observe that M (K 5 \e) and M (K 3,3 ) are contraction minors of PG(3, 2), whereas R 10 is a contraction minor of R17,so they aren t explicitly stated. Suppose M is non-regular. If M has no P 9 or P9 -minor, then by Theorem 3.1 M is isomorphic to F 7, F7, Z r, Zr, Z r \b r,orz r \c r,forsomer 4. Thus we may assume M has a P 9 or P9 -minor. Observe that, among the extensions of P 9, D 2 and D 3 have an M(K 5 \e)-minor and P 9 is a 3-decomposer for rank-4 binary 3-connected matroids without a D 2 -minor. This means P 9 is a 3-decomposer for D 1 (and consequently P9 is a 3-decomposer for D1 ). So D 1 is not explicitly listed. From Table 1 X 1, X 2 and X 3 have an M(K 5 \e)-minor, so no further rank 4 matroids are in the class. It follows from Theorem 3.1 (iv) that M is isomorphic to PG(3, 2) or R17 or its contraction minors. Acknowledgments A year and half ago, the first author submitted a research proposal outlining matroid representability problems. The unknown reviewer indicated we should tackle this problem. The authors thank the reviewer for highlighting it. The authors also thank the unknown referees for many helpful suggestions. Appendix All matroids mentioned below are binary and 3-connected. Table 1 gives all the extensions of P 9 upto the projective space PG(3, 2) using the matrix representation given in Sect. 1. Table 2 gives the single-element coextensions of P 9. Tables 3 and 4 give the single-element extensions of M(K 5 \e) and M (K 5 \e). A matrix representation for the graph K 5 \e is given below. For M (K 5 \e), the matrix representation used is in Sect K 5 \e = I Table 5 gives the single-extension coextensions for D 1 and D 2 using the matrix representation given in Sect. 3. Table 6 gives the single-element extensions of E 5

14 Table 1 Rank 4 extensions of P 9 Matroid Extension columns Name P 9 [1110] D 1 [1001][0101][0110], [1010] D 2 [0011] D 3 D 1 [0101][0110][1001][1010] X 1 [0011] X 2 D 2 [1010][1110] X 1 [0011][0101][0110] X 3 D 3 [1110] X 2 [0101][0110][1001][1010] X 3 X 1 [0011][0101][0110] Y 1 [1110] Y 2 X 2 [0101][0110][1001][1010] Y 1 X 3 [0101][0110][1010][1110] Y 1 Table 2 Single-element coextensions of P 9 Coextension rows Name [11000][11111] E 1 [11011][11100] E 2 [11001][11101] E 3 [01001][01010][01101][01110] E 4 [10001][10010][10101][10110] [01011][01100][10011][10100] E 5 [00101][00110] E 6 [00111] E 6 [00011] E 7 Table 3 Single-element extensions of M(K 5 \e) Extension columns Name [0101] K 5 [0111][1101][1111] D 2 [1011][1110] D 3 Table 4 Single-element extensions of M (K 5 \e) Extension columns Name [00101][00110][01100][01110] (K 5 \e) + edge [10100][10101] [01001][10010][11011] (K 3,3 ) [01010][01011][10001][10011] E 4 [11001][11010] [00111][][10111][11100] E 6 [11101][11110] [11111] E7

15 Table 5 Single-element coextensions of D 1 and D 2 Matroid Coextension rows Name Relevant minors D 1 [000011][000101][001010][001100][010010] E 1, E 2, E 3, E 4, E6 [010100][011011][011101][100010] [100100][101011][101101][110001] [110111][111000][111110] [000110] E 7 [000111][001110][010110][011001][100110] E 3, E 5, E6, E 7 [101001][110011][111010] [001001][0] E 2, E6 [001011][001101] E 2, E 3 E 5 [010001][010011][010101][010111][011000] E 2, E6 [011010][011100][0][10001] [100011][100101][100111][101000][101010] [101100][101110] [110000][110100][111101][111111] E 1, E 2 [110010][110110][111001][111011] E 2, E 3 D 2 [000011][000101][000110][0][100111][101000] A E 5 E 6 E 7 K 3,3 [011001] B E 5, K 3,3,R 10 [010111][110011][111010] C E 3 E 5 E 6, E 7 [000111] Z E 7, R 10 [001001][100100][101101] E 4 [001010][001100][100001][100010][101011][101110] E 4 [001011][001101][100101][100110][101001][101100] E 4 [001110][100011][101010] E 4 [010001][011000][011011][011101][110110][111001] E 4 [010010][010100][110000][110101][111100][111111] E 4 [010011][010101][110010][110111][111000][111011] E 4 [010110][011010][011100][1][110001][111110] E 4 Table 6 Single-element extensions of E 5 Extension columns Name Contraction-minor Deletion-minor [00101][00110][01011][01100] A D 2 E6, E 7, K 3,3 [10011] B D 2 K 3,3, R 10 [11001][11101] C D 2 E6, E 7, E 3 [00011][00111][01001][01101] D 2 E ][01110] D 2 E 4 [10001][10010][11011][11100] D 2 E 4 [10101][10110][11000][11111] D 2 E 4 using the matrix representation given in Sect. 1. Table 7 gives the single-element coextensions of A, B, C, and Z using the matrix representations given in Sect. 3. Table 8 gives all the extensions of E 5 up to R 17 with no M (K 5 \e)-minor. Table 9

16 Table 7 Single-element coextensions of A, B, C, andz Matroid Name Coextension row A coext 1 red [000011][000101][001010][011010][1][111001] coext 2 [000110][110011][110101] coext 3 [000111][101011][111011] coext 4 [001001][010110][1] coext 5 [001011][011011][100111] coext 6 [001100][011100][110000] coext 7 [001101][010010][010100][011101][101110][111000] coext 8 [001110][011000][101101][110010][110100][111101] coext 9 [0][011001][100011][100101][101010][111010] coext 10 [010001][100010][100100] coext 11 [010011][010101][100110] coext 12 [010111] coext 13 [100001][101000][111110] coext 14 [101001][110110][111111] B coext 1 [000011][000101][000110][001001][001010][0][010010][010100] [010111][011000][011011][0] coext 2 [000111][001011][010110][011010] coext 3 [001100][010001][011101] coext 4 [001101][001110][010011][010101][011001][011100] coext 5 [100001][100010][100100][101000][101101][101110][110000][110011] [110101][111001][111100][111111] coext 6 [100011][100101][101010][1][111000][111011] coext 7 [100110][101001][110010][110100][110111][111110] coext 8 [100111][101011][111010] C coext 1 [000011][000101][001001][0][010010][010100][011000][0] [100010][100100][101000][101110][110011][110101][111001][111111] coext 2 [000110][010111] coext 3 [000111][010110][100110][110111] coext 4 [001010][011011] coext 5 [001011][011010][101010][111011] coext 6 [001100][011101] coext 7 [001101][011100][101100][111101] coext 8 [001110][010011][010101][011001] coext 9 [010001] coext 10 [100001][110000] coext 11 [100011][100101][1][111000] coext 12 [100111] coext 13 [101001][110010][110100][111110] coext 14 [101011][111010]

17 Table 7 continued Matroid Name Coextension row Z coext 1 [000011][000101][001001][001110][010001] [011100][100001][100111][110010][111001] coext 2 [000110][001011][001101][010010][010101][011000][1][100010] [100100][101000][101110][110000][110111][111011][111100] coext 3 [000111][001010][001100][010011][010100][011001][0][100011] [100101][101001][1][110001][110110][111010][111101] coext 4 [010110][010111][011010][011011][101010][101011][101100][101101] [110100][110101][111110][111111] Table 8 All extensions of E 5 with no M (K 5 \e) -minor up to Matroid Extension column Name R 17 A [00110][01100][10011] D [01011] E [11001] F [11101] G B [00101][00110][01011][01101] D [11001][11101] E C [00101][01011][10011][11101] F [00110][01100] G D [01011][01100][10011] H [11001][11101] E E [00110][01100][10011] H [11001] J [11101] K F [00110][01100][10011][11101] I [01011] J G [00110][01100][10011][11001] I [01011] K H [01100][10011] L [11001][11101] M I [01011][01100][10011][11101] M J [00110][01100][10011][11101] M K [00110][01100][10011][11001] M L [10011] O [11001][11101] P M [01100][10011][11101] P O [11001][11101] Q P [10011][11101] Q Q [11101] R

18 Table 9 Single-element coextensions of X 1 and X 3 Matroid Name Coextension row X 1 coext 1 [ ][ ][ ][ ][ ][ ][ ] [ ] ][ ][ ][ ] coext 2 [000111][001011][ ][ ][10] [ ][ ][ ] coext 3 [00] coext 4 [ ][ ][ ][ ][ ][ ] [ ][ ] [ ][ ][01][10][ ][ ][ ] [ ][ ][ ][ ][ ][ ] [ ][ ][ ] coext 5 [ ][ ][ ][ ][ ][ ] [ ][ ][ ][ ][001][00] coext 6 [ ][ ][ ][00][ ][ ][ ] [ ][ ][ ][ ][ ][ ] [ ][ ][ ] [ ][ ][11][ ][ ][11] [ ][ ][ ] coext 7 [01][01][ ] coext 8 [ ][ ][ ][ ] ][10][110001] [ ] ][ ][ ][ ] coext 9 [100010][ ][ ][ ] coext 10 [ ][ ][ ][ ][ ][ ][ ] [ ][ ][ ][ ][ ] coext 11 [ ][ ][ ][ ] X 3 coext 1 [ ][ ][ ][ ][ ][ ][ ] [ ][ ][ ][ ] coext 2 [ ][ ][ ][00][ ][ ][ ] [11][ ][ ][ ][ ] coext 3 [ ][ ][ ][01][ ][ ] coext 4 [ ][01][ ][ ] coext 5 [00] coext 6 [ ][ ][ ] coext 7 [ ][ ][ ][ ][11][ ] coext 8 [ ][ ][ ][00][ ][ ][ ] [10][ ][ ][ ][ ] coext 9 [ ][ ][ ][ ][ ][01][ ] [ ][ ][ ][ ][ ] coext 10 [ ][ ][ ][ ][ ][ ][ ] [ ][ ][ ][ ][ ] coext 11 [ ][ ][ ][ ][ ][ ] ] ][ [ ][ ] coext 12 [ ][ ][ ][ ] ][ ] coext 13 [ ][ ][ ]

19 Table 9 continued Matroid Name Coextension row coext 14 [ ][ ][ ] coext 15 [ ][ ][ ][11] coext 16 [ ][ ][ ][ ] coext 17 [ ][ ][ ] coext 18 [10] gives the single-element coextensions of X 1 and X 3 using the matrix representations given in Sect. 3. Type II and III rows are marked in red in Tables 7 and 9. References 1. Dirac, G.A.: Some results concerning the structure of graphs. Canad. Math. Bull. 6, (1963) 2. Kingan, S.R.: Binary matroids without prisms, prism duals, and cubes. Discret. Math. 152, (1996) 3. Kingan, S.R., Lemos M.: Strong Splitter Theorem. Ann. Combinatorics (to appear) 4. Mayhew, D., Royle, G., Whittle G.: The internally 4-connected binary matroids with no M(K 3,3 )-minor, Memoirs of the American Mathematical Society, vol American Mathematical Society, Providence (2011) 5. Mayhew, D., Royle, G.: The internally 4-connected binary matroids with no M(K 5 \e)-minor. Siam J. Discret. Math. 26, (2012) 6. Oxley, J.G.: The binary matroids with no 4-wheel minor. Trans. Amer. Math. Soc. 154, (1987) 7. Oxley, J.G.: Matroid Theory (2011), 2nd edn. Oxford University Press, New York (1992) 8. Robertson, N., Seymour, P.D.: Generalizing Kuratowski s Theorem. Congressus Numerantium 45, (1984) 9. Seymour, P.D.: Decomposition of regular matroids. J. Combin. Theory Ser. B 28, (1980)