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JOURNAL OF COMBINATORIAL THEORY @) 22, 289-295 (1977)

Note

Note on the Production of

P. D. SEYMOUR*

Department of Pure Mathematics, University College, Singleton Park, Swansw, SA2 8PP, South WaE2s Communicated by the Managing Editors Received April 1975

The circuits containing some fixed element of a connected (such a collection is called a port) provide a matroid generalization of the “‘path collec- tions” of graphs. In this note we show how to translate forbidden minor theorems in matroid theory into results about ports - and we find that many theorems are strengthened by such translation. Those collections of sets which care “path collections” of graphs may then be characterized.

is a matroid, let E(M) be the set of elements o the of - Z and with circu - Z. and M/Z is (M*\Z)*. The binary relation “x = y or there is containing both x and y” may be shown to be an equivalence the equivalence classes are called elementary se~~r~~Q~~ ementary separators is a separator. Z is an elementary separator of is connected if it has at most one is a separator if and only if CWWO. s a collection of sets such that A, Q AZ for A, , A, E L. UL is I.) (A: A E L). If 9 is an element of a matroi (C - (J2j: !2 E 62 E %?I, where VT is the collection clutter; such a clutter is called aport [5]. X2, , \Z,j%,) is the collection of minimal members of(A - Z%: A E A n Z, = B:>. This suggests that if E is a chnter and ZI ) Z, C UL are disjoint, we define L\Z,/Z, to be the collection of minimal members of (A - Z,: A E L, A R Z, = a) and develop the notion or” clutter nzino~ (see 16, g] for more detail). We find that ports can be ~~aracter~~ed among

* Present address: Merton College, Oxford. 289 Copyright Q 1977 by Academic Press, Inc. &I rights of reproduction in any form reserved. 290 P. D. SEYMOUR clutters by a forbidden minor condition [6]. Lehman [4] showed that if M is connected, O(M) determines M uniquely, so the problem arises, which properties of Q(M) give rise to which properties of M? (See [9].) In this note, we study how to characterize ports of with no matroid minor in a set K by a forbidden clutter minor condition. This enables us to characterize (for example) the ports of binary, ternary (representable over GF(3)), regular, and graphic matroids. We find that several familiar classes K of connected matroids have the property that if Q is an element of a connected matroid M with a minor in K, then M has a minor in K using 9. If K is a class of connected matroids with this property, we say that K is rounded. For instance, (U,z) is rounded [l]. It is clear that if K is rounded and $2 is an element of a connected matroid M, then M has a matroid minor in K if and only if Q(M) has a clutter minor in the collection of ports of members of K. Our problem thus reduces to, given a set K of connected matroids, find a rounded set K+ including K such that each member of K+ has a minor in K. If we can do this, then, for an element 1;2 of a connected matroid M, the following are equivalent:

(i) M has a matroid minor in K; (ii) M has a matroid minor in K+; (iii) Q(M) has a clutter minor which is a port of a member of KT.

To keep condition (iii) as simple as possible, it is desirable that K+ should be as small as possible. If K is a set of connected matroids, we define K+ = K u (N: N is connected and there exists Q E E(N) such that N\(Q) E K or N/(e) E K, but N has no minor in K using Sz, and N has no connected proper minor using Q with a minor in K). K+ will satisfy our requirements. I am indebted to the referee and to L. Schrijver for pointing out the following lemma, which greatly improves on my silly original method of Proof.

LEMMA. If N is a connected minor of a connected matroid M, and e E E(M) - E(N), then at least one of M\(e), M/(e) is connected and has N as a minor.

Proof. We may assume by duality that N is a minor of M\(e). If M\(e) is connected, we are done. Otherwise (see [12]) M/(e) is connected, and we require to show that N is a minor of M/(e} as well. N is connected, and so is a minor of some component M’ of M\(e). Put E = E(M), E1 = E(M’), E, = E - (El u {e>). M is connected, so r(E, u {e)) + r(E2) > r(E), but 4%) + 4.G = r(E - (4) < r(M) since E, is a separator of M\(e); so r(E, u (e>) > r(E,) and e is a coloop of M\E, . Thus M’ = M\E,\(e} = M\E,/{e) = M/{e)\E2, and N is a minor of M/(e) as required. PRODUCTION OF MATROD MINORS 291

THEOREM. jf K- is a set of connected matvoids, K+ is rounded.

PUOQJ Suppose that Q E E(M) and is connected and has a minor in K+ (and hence in K). Choose a minimal connected mi with a minor in K. Let R/H’ E K be a minor of N. Hf .G’ E so suppose that Q # E&l’). If e E E(N) - ) then by the lemma either is connected and has the mino ‘; so by the minima&y of ‘> u (Q}. Thus N\(s2} E K or N E K; if N has a minor in using J2 we are done, and if not, then N E KA, as required. We can also show that if J 2 K is rounded, and each minor in K, then K+ 1 J, and so K+ is as small as possible. If Kis a set of connected matroids, let P(K) be {Q Psz( E K- - K and Q E E(M) such that Then we have (1) if 29 is an element of a connected matroid M, then M has a ma&id minor in Kif and only if Q(M) has a clutter minor in P(K); (2) if no member of K has another member of M as a matroid minor, then no member of P(K) has another member of P(K) as a clutter minor. The proofs are straightforward and we omit them. It follows that if K is the set of forbidden matroid minors for some propercy, then P(K) is the corre- sponding set of forbidden clutter minors. Now we are in a position to apply the theorem. We take an interesting matroid property preserved under matroid minors, find the set K of forbidden minors for the property, evaluate K+ and then P(K). We find that, for some unexplained reason, K+ = K quite frequently; if so, we have a direct strength erring of the original forbidden minor theorem. For instance, it follows from the next result and Tutte’s characterization of regular matroids [I I]> that if Q is an element of a connected matroid I?4 which is not regular, then has a minor U421P or F* using .Q.

(iv) ~1u=(~~z,F,F”,~*(K,),~*(K3,3))thelzK’=Kv(~*(K,,,i-e):, where K3,, + e is K3,3 extended by one new edge joining two ~on~dj~~e~t vertices (graphic 1131). (v) .ivf K = { U5*, Us3, F, F*] then K+ = K (representable over GF(3) P, m 292 P. D. SEYMOUR

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(1 "I ‘I ‘I '0) : 6 tr ‘0‘0 “I ‘01: 8 (1 ‘T ‘0 ‘0 '0) : L to 'I ‘I ‘0‘0) : 9 to‘o'I‘1To~ :s tL1 (0'1 ‘I ‘I "0 (I ‘o‘o‘o‘r> : P Co"1 "0'0‘1~ : E {i> to '0 "I ‘0 'I) C0‘0‘1 ‘0 "1) : z (O"O‘O'l‘T~ : I 294 P. D. SEYMOUR

ProoJ Define K* = {M*: M E K}. For each of these sets K, we must check that if .Q is an element of a connected matroid M and M\(Q) E K (or M\{Q] E K*) then M has a minor in K (or, respectively, in K*) using 52. A2 is not a loop or coloop because M is connected; if .Q is parallel to e, say then M\(e) is isomorphic to M\{Qj, and so M\{e} E K on K* as required; and each member of K, K* is geometric, so M\{Q} is geometric. Thus we assume may that M is geometric. (i) It follows that M = U,Z; so M\(e) = Ua2 E K = K* for any element e f L?, as required.

In cases (ii), (iii), (iv), Uq2 E K, I(*, and so if M is not binary then from case (i) M has a minor in K and K* using 8 (namely, Uaz); we may thus assume that M is binary. To find all binary geometric matroids M with an element 1;2 such that M\(Q) E K (or K*), we take a binary representation of each member of K (or K*) and add single new points v of the vector space in all possible ways. We need consider only those v different from each point used by the representation, because M is geometric. In the representations of members of K (or K*) given in Table I, each vector used has an even number of nonzero coordinates-it suffices to assume that v also has an even number of nonzero coordinates, for otherwise v represents a coloop. Table I lists all nonzero v satisfying these conditions up to permutation of the coordinate places inducing automorphisms of M\{Q}, and gives a corresponding minor of M using Q in the set asserted to be K+ (or K*+ = K+*). I have given F or F* minors whenever possible; A!‘*(K3,J is given only when M is regular. This table deals with (ii), (iii), (iv). For case (v) we must consider the nonbinary extensions of F*-there are a lot of these, and they are not conveniently enumerable, so we omit the details. We can use case (iv) to characterize those clutters which are path collections of graphs, viz:

THEOREM. If L is a clutter, then there is an undirected graph G and distinct vertices u, v of G such that L is the collection of edge sets of the simple paths between u and v, if and only if L has no clutter minor in the list:

(9 {O, 21, $2, 31, (3,411; (ii) ((1, 2,..., s>> U ((0, i]: i = 1 **a s} (s > 2); (iii) Q(F) for some element Q; (iv) J&F*) for some element J& (v) Q(JJ!*(K,)) for some element Sz; (vi) Q(A*(K,,,)) for some element Qn; (vii) Q(J~‘*(&, + e)) where 8 is the new edge. PRODUCTION OF MATROID M~~O~~ 295

ProoJ There exists a graph G and vertices U, u as in the theorem if and only if L = &2(M) for some M (remove the edge 92 to produce G). L = C!(M) for some connected no clutter minor (i) or (ii) [8]. M is graphic if and o minor in K = {U4”, F, F*, A?‘*(&,), A?*(&J} [II, 131, and B(K) is the set of clutters in (iii) .‘. (vii) togeiher with ((1, 21, (2, 3>>(3, I>>, which iies in (ii). This completes the proof.

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