An obstacle to a decomposition theorem for near-regular matroids
aa r X i v : . [ m a t h . C O ] J a n An obstacle to a decomposition theorem fornear-regular matroids ∗ Dillon Mayhew † Geoff Whittle † Stefan H. M. van Zwam ‡ June 7, 2018
Abstract
Seymour’s Decomposition Theorem for regular matroids states thatany matroid representable over both GF(2) and GF(3) can be obtainedfrom matroids that are graphic, cographic, or isomorphic to R by1-, 2-, and 3-sums. It is hoped that similar characterizations hold forother classes of matroids, notably for the class of near-regular matroids.Suppose that all near-regular matroids can be obtained from matroidsthat belong to a few basic classes through k -sums. Also suppose thatthese basic classes are such that, whenever a class contains all graphicmatroids, it does not contain all cographic matroids. We show that inthat case 3-sums will not suffice. A regular matroid is a matroid representable over every field. Much isknown about this class, the deepest result being Seymour’s Decompo-sition Theorem:
Theorem 1.1 (Seymour [16]) . Let M be a regular matroid. Then M can be obtained from matroids that are graphic, cographic, or equal to R through -, -, and -sums. A class C of matroids is polynomial-time recognizable if there existsan algorithm that decides, for any matroid M , in time f ( | E ( M ) | , τ )whether or not M ∈ C , where τ is the time of one rank evaluation, and f ( x, y ) a polynomial. Seymour [17] showed that the class of graphic ∗ Parts of this research have appeared in the third author’s PhD thesis [24]. Theresearch of all authors was partially supported by a grant from the Marsden Fund of NewZealand. The first author was also supported by a FRST Science & Technology post-doctoral fellowship. The third author was also supported by NWO, grant 613.000.561. † School of Mathematics, Statistics and Operations Research, Victoria Uni-versity of Wellington, New Zealand. E-mail:
[email protected] , [email protected] ‡ Centrum Wiskunde en Informatica, Postbus 94079, 1090 GB Amsterdam, The Nether-lands. E-mail:
[email protected] atroids is polynomial-time recognizable. Also every finite class ispolynomial-time recognizable. Using these facts Truemper [18] (seealso Schrijver [14, Chapter 20]) showed the following: Theorem 1.2.
The class of regular matroids is polynomial-time rec-ognizable. A near-regular matroid is a matroid representable over every field,except possibly GF(2). Near-regular matroids were introduced byWhittle [19, 20]. The following is one of his results: Theorem 1.3 (Whittle [20]) . Let M be a matroid. The following areequivalent:i. M is representable over GF(3) , GF(4) , and
GF(5) ;ii. M is representable over Q ( α ) by a totally near-unimodular ma-trix;iii. M is near-regular. In this theorem α is an indeterminate. A totally near-unimodularmatrix is a matrix over Q ( α ) such that the determinant of every squaresubmatrix is either zero or equal to ( − s α i (1 − α ) j for some s, i, j ∈ Z .Whittle [20, 21] wondered if an analogue of Theorem 1.1 would holdfor the class of near-regular matroids. The following conjecture wasmade: Conjecture 1.4.
Let M be a near-regular matroid. Then M can beobtained from matroids that are signed-graphic, their duals, or membersof some finite set through -, -, and -sums. A matroid is signed-graphic if it can be represented by a GF(3)-matrix with at most two nonzero entries in each column (see Za-slavsky [22, 23] for more on these matroids). One difference with theregular case is that not every signed-graphic matroid is near-regular.Several people have made an effort to understand the structure ofnear-regular matroids. Oxley et al. [7] studied maximum-sized near-regular matroids. Hlinˇen´y [5] and Pendavingh [10] have both writtensoftware to investigate all 3-connected near-regular matroids up to acertain size. Pagano [9] studied signed-graphic near-regular matroids,and Pendavingh and Van Zwam [11] studied a closely related class ofmatroids which they call near-regular-graphic.Despite these efforts, an analogue to Theorem 1.1 is still not insight. In this paper we record an obstacle we found, that will haveto be taken into account in any structure theorem. Our result is thefollowing:
Theorem 1.5.
Let G , G be graphs. There exists an internally 4-connected near-regular matroid M having both M ( G ) and M ( G ) ∗ asa minor. From this, and the fact that not all cographic matroids are signed-graphic, it follows that Conjecture 1.4 is false. More generally, supposewe want to find a decomposition theorem for near-regular matroids,such that each basic class that contains all graphic matroids, does ot contain all cographic matroids. Theorem 1.5 implies that such acharacterization must employ at least 4-sums.The paper is organized as follows. In Section 2 we give some pre-liminary definitions. In Section 3 we prove a lemma that shows howgeneralized parallel connection can preserve representability over a par-tial field. In Section 4 we prove Theorem 1.5. We conclude in Section 5with some updated conjectures.Throughout this paper we assume familiarity with matroid theoryas set out in Oxley [8]. In addition to the usual definitions of connectivity and separations (seeOxley [8, Chapter 8]) we say a partition (
A, B ) of the ground set of amatroid is k -separating if rk M ( A ) + rk M ( B ) − rk( M ) < k . Recall that( A, B ) is a k -separation if it is k -separating and min {| A | , | B |} ≥ k . Definition 2.1.
A matroid is internally 4-connected if it is 3-connectedand min {| X | , | Y |} = 3 for every 3-separation ( X, Y ).This notion of connectivity is useful in our context. For instance,Theorem 1.1 can be rephrased as follows:
Theorem 2.2.
Let M be an internally 4-connected regular matroid.Then M is graphic, cographic, or equal to R . Intuitively, separations (
X, Y ) where both | X | and | Y | are bigshould give rise to a decomposition into smaller matroids. Definition 2.3.
Let M be a matroid, and N a minor of M . Let( X ′ , Y ′ ) be a k -separation of N . We say that ( X ′ , Y ′ ) is induced in M if M has a k -separation ( X, Y ) such that X ′ ⊆ X and Y ′ ⊆ Y .At several points we will use the following easy fact: Lemma 2.4.
Let M be a matroid, let N be a minor of M , and let ( A, B ) be a k -separating partition of E ( M ) . Then ( A ∩ E ( N ) , B ∩ E ( N )) is k -separating in N . Note that ( A ∩ E ( N ) , B ∩ E ( N )) need not be exactly k -separating. Our main tool in the proof of Theorem 1.5 is useful outside the scopeof this paper. Hence we have stated it in the general framework ofpartial fields. For that purpose we need a few definitions. More on thetheory of partial fields can be found in Semple and Whittle [15] and inPendavingh and Van Zwam [13, 12].
Definition 2.5. A partial field is a pair ( R, G ), where R is a commu-tative ring with identity, and G is a subgroup of the group of units of R such that − ∈ G . or example, the near-regular partial field is ( Q ( α ) , h− , α, − α i ),where h S i denotes the multiplicative group generated by S . For P =( R, G ), we abbreviate p ∈ G ∪ { } to p ∈ P .We will adopt the convention that matrices have labelled rows andcolumns, so an X × Y matrix A is a matrix whose rows are labelled bythe (ordered) set X and whose columns are labelled by the (ordered)set Y . The identity matrix with rows and columns labelled by X willbe denoted by I X . We will omit the subscript if it can be deducedfrom the context.Let A be an X × Y matrix. If X ′ ⊆ X and Y ′ ⊆ Y then we denotethe submatrix of A indexed by X ′ and Y ′ by A [ X ′ , Y ′ ]. If Z ⊆ X ∪ Y then we write A [ Z ] := A [ X ∩ Z, Y ∩ Z ]. If A is an X × Y matrix, where X ∩ Y = ∅ , then we denote by [ I A ] the X × ( X ∪ Y ) matrix obtainedfrom A by prepending the identity matrix I X . Definition 2.6.
Let P := ( R, G ) be a partial field, and let A be amatrix with entries in R . Then A is a P -matrix if, for every squaresubmatrix A ′ of A , either det( A ′ ) = 0 or det( A ′ ) ∈ G . Theorem 2.7.
Let P be a partial field, let A be an X × Y P -matrixfor disjoint sets X and Y , let E := X ∪ Y , and let A ′ := [ I A ] . If B = { B ⊆ E : | B | = | X | , det( A ′ [ X, B ]) = 0 } , then B is the set of basesof a matroid. We denote this matroid by M [ I A ]. Let A be an X × Y P -matrix. Then X is a basis of M [ I A ]. We say that X is the displayed basis. Pivoting in the matrix allows us to changethe basis that is displayed. Roughly speaking a pivot in A consists ofrow reduction applied to [ I A ], followed by a column exchange. Theprecise definition is as follows:
Definition 2.8.
Let A be an X × Y matrix over a ring R , and let x ∈ X, y ∈ Y be such that A xy ∈ R ∗ . Then A xy is the ( X − x ) ∪ y × ( Y − y ) ∪ x matrix with entries( A xy ) uv = ( A xy ) − if uv = yx ( A xy ) − A xv if u = y, v = x − A uy ( A xy ) − if v = x, u = yA uv − A uy ( A xy ) − A xv otherwise.We say that A xy was obtained from A by pivoting . Slightly lessopaquely, if A = " y Y ′ x a c X ′ b D hen A xy = " x Y ′ y a − a − c X ′ − ba − D − ba − c . As Semple and Whittle[15] proved, pivoting maps P -matrices to P -matrices: Proposition 2.9.
Let A be an X × Y P -matrix, and let x ∈ X, y ∈ Y besuch that A xy = 0 . Then A xy is a P -matrix, and M [ I A ] = M [ I A xy ] . Semple and Whittle also showed that pivots can be used to computedeterminants of P -matrices: Lemma 2.10.
Let P be a partial field, and let A be an X × Y P -matrixwith | X | = | Y | . If x ∈ X, y ∈ Y is such that A xy = 0 then det( A ) = ( − x + y A xy det( A xy [ X − x, Y − y ]) . Recall the generalized parallel connection of two matroids M , M along a common restriction N , denoted by P N ( M , M ). This con-struction was introduced by Brylawski [1] (see also Oxley [8, Sec-tion 12.4]). Brylawski proved that representability over a field canbe preserved under generalized parallel connection, provided that therepresentations of the common minor are identical. Lee [6] general-ized Brylawski’s result to matroids representable over a field such thatall subdeterminants are in a multiplicatively closed set. We generalizeBrylawski’s result further to matroids representable over a partial field,as follows. Theorem 3.1.
Suppose A , A are P -matrices with the followingstructure: A = (cid:20) Y YX D ′ X D D X (cid:21) , A = (cid:20) Y Y X D X D X D ′ (cid:21) , where X, Y, X , Y , X , Y are pairwise disjoint sets. If X ∪ Y is amodular flat of M [ I A ] then A := Y Y Y X D ′ X D D X D X D ′ is a P -matrix. Moreover, if M = M [ I A ] and M = M [ I A ] , then M [ I A ] = P N ( M , M ) , where N = M [ I D X ] . he main difficulty is to show that A is a P -matrix. To provethis we will use a result known as the modular short-circuit axiom [1,Theorem 3.11]. We use Oxley’s formulation [8, Theorem 6.9.9], andrefer to that book for a proof. Lemma 3.2.
Let M be a matroid and X ⊆ E nonempty. The follow-ing statements are equivalent:i. X is a modular flat of M ;ii. For every circuit C such that C − X = ∅ , there is an element x ∈ X such that ( C − X ) ∪ x is dependent.iii. For every circuit C , and for every e ∈ C − X , there are an f ∈ X and a circuit C ′ such that e ∈ C ′ and C ′ ⊆ ( C − X ) ∪ f . The following is an extension of Proposition 4.1.2 in [1] to par-tial fields. Note that Brylawski proves an “if and only if” statement,whereas we only state the “only if” direction.
Lemma 3.3.
Let M = ( E, I ) be a matroid, and X a modular flatof M . Suppose B X is a basis for M | X , and B ⊇ B X a basis of M .Suppose A is a B × ( E − B ) P -matrix such that M = M [ I A ] . Thenevery column of A [ B X , E − ( B ∪ X )] is a P -multiple of a column of [ I A [ B X , X − B ]] .Proof of Lemma 3.3. Let M , X , B X , B , A be as in the lemma, so A = (cid:20) E − ( B ∪ X ) X − BB − B X D ′ B X D D B X (cid:21) . Let v ∈ E − ( B ∪ X ), and let C be the B -fundamental circuit containing v . If C ∩ X = ∅ then D [ B X , v ] is an all-zero vector and the result holds,so assume B X ∩ C = ∅ . By Lemma 3.2( iii ) there are an x ∈ X and acircuit C ′ with v ∈ C ′ and C ′ ⊆ ( C − X ) ∪ x .Let M ′ := M/ ( B − B X ). Then C ′ ∩ E ( M ′ ) = { v, x } is a circuitof M ′ . Hence all 2 × I A ][ B X , { v, x } ] have tobe 0, which implies that A [ B X , v ] is the all-zero vector or parallel to[ I A ][ B X , x ]. Proof of Theorem 3.1.
Let A , A , A be as in the theorem, and define E := X ∪ X ∪ X ∪ Y ∪ Y ∪ Y . Suppose there exists a Z ⊆ E such that A [ Z ] is square, yet det( A [ Z ]) P . Assume A , A , A, Z were chosenso that | Z | is as small as possible.If Z ⊆ X i ∪ Y i ∪ X ∪ Y for some i ∈ { , } then A [ Z ] is a submatrixof A i , a contradiction. Therefore we may assume that Z meets both X ∪ Y and X ∪ Y . We may also assume that A [ Z ] contains norow or column with only zero entries, so either there are x ∈ X ∩ Z , y ∈ Y ∩ Z with A xy = 0 or x ∈ X ∩ Z , y ∈ Y ∩ Z with A xy = 0.In the former case, pivoting over xy leaves D X , D , and D ′ un-changed, yet by Lemma 2.10 det( A [ Z ]) ∈ P if and only if det( A xy [ Z −{ x, y } ]) ∈ P . This contradicts minimality of | Z | . Therefore Z ∩ X = ∅ .Similarly, Z ∩ X = ∅ . efine X ′ := Z ∩ X . Now pick some y ∈ Y . Since A [ X ′ , Y ∪ Y ] isobtained from A [ X, Y ∪ Y ] by deleting rows, it follows from Lemma 3.3,applied to M [ I A ], that the column A [ X ′ , y ] is either a unit vector (i.e.a column of an identity matrix) or parallel to A [ X ′ , y ′ ] for some y ′ ∈ Y .In the first case, Lemma 2.10 implies again that det( A [ Z ]) ∈ P if andonly if det( A [ Z − { x, y } ]) ∈ P , contradicting minimality of | Z | . In thesecond case, if y ′ ∈ Z then det( A [ Z ]) = 0. Otherwise we can replace y by y ′ without changing det( A [ Z ]) (up to possible multiplication withsome nonzero p ∈ P ). It follows that det( A [ Z ]) = p ′ det( A [ Z ′ ]), where Z ′ ⊆ X ∪ Y ∪ Y , and p ′ ∈ P − { } . But det( A [ Z ′ ]) ∈ P , so alsodet( A [ Z ]) ∈ P , a contradiction.It remains to prove that M [ I A ] = P N ( M , M ). Suppose P =( R, G ), and let I be a maximal ideal of R . Let ϕ : R → R/I bethe canonical ring homomorphism. For a square P -matrix D we havedet( D ) = 0 if and only if det( ϕ ( D )) = 0. Hence M [ I A ] = M [ I ϕ ( A )].But R/I is a field, so the result now follows directly from Brylawski’soriginal theorem.The special cases X = ∅ and X = { p } were previously proven bySemple and Whittle [15]. The core of the proof of Theorem 1.5 will be a special matroid M := M [ I A ], where A = d e f a b − α c α − α − . (1) Lemma 4.1.
The following hold:i. M is near-regular;ii. M is internally 4-connected;iii. M is self-dual;iv. M \{ , , , , , } ∼ = M ( K ) ;v. M / { a, b, c, d, e, f } ∼ = M ( K ) ;vi. No triad of M \{ , , , , , } is a triad of M . We will omit the proofs, each of which boils down to a finite casecheck that is easily done on a computer and not too onerous by hand.Specifically, for the first property one can either verify that A istotally near-unimodular, or that M contains none of the excludedminors for near-regular matroids (see Hall et al. [4]). The latter ap-proach is facilitated by observing that M is the signed-graphic ma-troid associated with the signed graph illustrated in Figure 1. That d ce af b
61 53 4
Figure 1: Signed-graphic representation of M . Negative edges are dashed;positive edges are solid. graph can also be used to verify ( ii ), by examining all edge-partitions( A, B ) that meet in two or three vertices. The remaining propertiesare readily extracted from the matrix A .We will use the M ( K )-restriction to create the generalized parallelconnection of M with M ( K n ). The following is well-known, but wewill include the short proof. Lemma 4.2.
The matroid M ( K n ) is internally 4-connected.Proof. Fix an integer n , and suppose ( A, B ) is a 3-separation of M ( K n )with | A | , | B | ≥
4. It follows that n ≥
5. Assume that rk( A ) ≥ rk( B ).Note that cl( A ) and cl( B ) induce complete subgraphs of K n , and thatthese subgraphs meet in at most three vertices. It follows that, forsome vertex v of K n , all edges incident with v are in A , or all edgesare in B . Assume the former. Then cl( A ) = E ( K n ), and thereforerk( A ) = n −
1, and rk( B ) = 2. But then B is a subset of a triangle of K n , a contradiction.We need to show that in forming the generalized parallel connectionwe do not introduce unwanted 3-separations. The following lemmatakes care of this. Lemma 4.3.
Let M = M ( K n ) for some n ≥ , and M an internally4-connected matroid such that there is a set X = E ( M ) ∩ E ( M ) with N := M | X = M | X ∼ = M ( K ) . Then M := P N ( M , M ) is a well-defined matroid. If no triad of N is a triad of M then M is internally4-connected.Proof. It is well-known (see [8, Page 236]) that N is a modular flatof M . Hence M = P N ( M , M ) is well-defined. It remains to provethat M is internally 4-connected. Suppose not. M is obviously con-nected. Suppose ( A, B ) is a 2-separation of M . By relabelling wemay assume | A ∩ E ( M ) | ≥ | B ∩ E ( M ) | . By Lemma 2.4 we havethat ( A ∩ E ( M ) , B ∩ E ( M )) is 2-separating in M (since M is arestriction of M ). But M is 3-connected, so | B ∩ E ( M ) | ≤
1. Sim-ilarly we have either | A ∩ E ( M ) | ≤ | B ∩ E ( M ) | ≤
1. Since | E ( M ) ∩ E ( M ) | = 6, the latter must hold. Hence B = { e, f } for some e ∈ E ( M ) − E ( N ) and f ∈ E ( M ) − E ( N ). Since E ( M ) and E ( M ) re flats of M , we have rk M ( { e, f } ) = 2. Moreover e ∈ cl M ( E ( M ) − e )and f ∈ cl M ( E ( M ) − f ), so { e, f } ⊆ cl M ( A ). But thenrk M ( A ) + rk M ( B ) − rk( M ) = rk M ( B ) = 2 , (2)contradicting the fact that ( A, B ) is a 2-separation.Next suppose that (
A, B ) is a 3-separation of M with | A | ≥ | B | ≥
4. By relabelling we may assume | A ∩ E ( M ) | ≥ | B ∩ E ( M ) | . ByLemma 2.4 again, ( A ∩ E ( M ) , B ∩ E ( M )) is 3-separating in M . Since M is internally 4-connected, | B ∩ E ( M ) | ≤
3. Define T := B ∩ E ( M ).We will show that T ⊆ cl M ( B − T ). Since M has no cocircuits ofsize less than 4, we have T ⊆ cl M ( A ). Thereforerk M ( A ∪ T ) + rk M ( B − T ) − rk( M ) = rk M ( A ) + rk M ( B − T ) − rk( M ) ≤ rk M ( A ) + rk M ( B ) − rk( M ) = 2 . (3)If | B − T | ≥ M ( B ) = rk M ( B − T ). If | B − T | = 1 then rk M ( B − T ) = 1and we must have rk M ( B ) = 2. In that case T is a triangle of M andsome element e ∈ E ( M ) − E ( M ) is in the closure of T . But no suchelement e exists, since E ( M ) is a flat of M .Note that B − T ⊆ E ( M ). Since T ⊆ cl M ( B − T ) and E ( M )is a flat of M , we have that T ⊆ E ( M ). Hence T ⊆ E ( N ), and B ∩ E ( M ) = B . Since ( A ∩ E ( M ) , B ∩ E ( M )) is 3-separating and | B ∩ E ( M ) | = | B | ≥
4, we have | A ∩ E ( M ) | ≤
3. But | B ∩ E ( M ) | ≤ E ( N ) − B ⊆ A ∩ E ( M ), from which it follows that | A ∩ E ( M ) | ≥ N is a triad of M , we must have that A ∩ E ( M )is a triangle of M . Hence B ∩ E ( N ) is a triad of N . Now consider ( A ∩ E ( M ) , B ∩ E ( M )) again. This partition of M must be 3-separating,but B ∩ E ( M ) is not a triangle of M , and M has no 3-elementcocircuits. This contradiction completes the proof. Proof of Theorem 1.5.
It suffices to prove the theorem for G = G = K n , where n ≥
5. Label the edges of some K -restriction N of G by { a, b, c, d, e, f } , and define M ′ := ( P N ( M ( G ) , M )) ∗ . (4)By Theorem 3.1, M ′ is near-regular, and by Lemma 4.3, M ′ is inter-nally 4-connected.Note that we still have M ′ |{ , , , , , } ∼ = M ( K ). Label theedges of some K -restriction N of G by { , , , , , } , and define M := P N ( M ( G ) , M ′ ) . (5)By Theorem 3.1, M is near-regular, and by Lemma 4.3, M is internally4-connected. The result follows.Matroid M was found while studying the 3-separations of R .The unique 3-separation ( X, Y ) of R with | X | = | Y | = 6 is induced n the class of regular matroids. Pendavingh and Van Zwam had found,using a computer search for blocking sequences, that it is not inducedin the class of near-regular matroids.Unlike R and R in Seymour’s work, the matroid M by itselfis quite inconspicuous. A natural class of near-regular matroids is theclass of near-regular signed-graphic matroids. As indicated earlier, M is a member of this class (see Figure 1). The K -restriction is readilyidentified. M is self-dual and has an automorphism group of size 6,generated by ( c, e )( d, f )(1 , ,
6) and ( a, d )( b, e )(1 , , While Theorem 1.5 is a bit of a setback, we remain hopeful that a satis-factory decomposition theory for near-regular matroids can be found.First of all, the construction in Section 4 employs only graphic ma-troids. In fact, it seems difficult to extend the M ( G )-restriction ofthe 4-sum to some strictly near-regular matroid. The proof of Theo-rem 1.5 suggests the following construction: Definition 5.1.
Let M , M be matroids such that E ( M ) ∩ E ( M ) = X , N := M | X = M | X ∼ = M ( K k ), and M is graphic. Then the graph k -clique sum of M and M is P N ( M , M ) \ X .Now we offer the following update of Conjecture 1.4: Conjecture 5.2.
Let M be a near-regular matroid. Then M can beobtained from matroids that are signed-graphic, are the dual of a signed-graphic matroid, or are members of a finite set C , by applying thefollowing operations:i. 1-, 2-, and 3-sums;ii. Graph k -clique sums and their duals, where k ≤ . Note that the work of Geelen et al. [3], when finished, should implya decomposition into parts that are bounded-rank perturbations ofsigned-graphic matroids and their duals. However, the bounds theyrequire on connectivity are huge. Conjecture 5.2 expresses our hopethat for near-regular matroids specialized methods will give much morerefined results.As noted in the introduction, Seymour’s Decomposition Theoremis not the only ingredient in the proof of Theorem 1.2. Another re-quirement is that the basic classes can be recognized in polynomialtime. The following result suggests that this may not hold for thebasic classes of near-regular matroids:
Theorem 5.3.
Let M be a signed-graphic matroid. Let N be a matroidon E ( M ) given by a rank oracle. It is not possible to decide if M = N using a polynomial number of rank evaluations. A matroid is dyadic if it is representable over GF( p ) for all primes p >
2. Since all signed-graphic matroids are dyadic (which was firstobserved by Dowling [2]), this in turn implies that dyadic matroids arenot polynomial-time recognizable. proof of Theorem 5.3, analogous to the proof by Seymour [17]that binary matroids are not polynomial-time recognizable, was foundby Jim Geelen and, independently, by the first author. It involvesternary swirls, which have a number of circuit-hyperplanes that is ex-ponential in the rank. To test if the matroid under consideration isreally the ternary swirl, all these circuit-hyperplanes have to be exam-ined, since relaxing any one of them again yields a matroid.However, this family of signed-graphic matroids is not near-regularfor all ranks greater than 3. Hence the complexity of recognizing near-regular signed-graphic matroids is still open. The techniques used bySeymour [17] do not seem to extend, but perhaps some new idea canyield a proof of the following conjecture: Conjecture 5.4.
Let C be the class of near-regular signed-graphic ma-troids. Then C is polynomial-time recognizable. In fact, we still have some hope for the following:
Conjecture 5.5.
The class of near-regular matroids is polynomial-time recognizable.
Acknowledgements
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