Recognizing Even-Cycle and Even-Cut Matroids

Abstract

Even-cycle matroids are elementary lifts of graphic matroids. Even-cut matroids are elementary lifts of cographic matroids. We give a polynomial time algorithm to check if a binary matroid is an even-cycle matroid. We also give a polynomial time algorithm to check if a binary matroid is an even-cut matroid. These algorithms rely on structural properties of the class of pinch-graphic matroids.

Supported by NSERC grant 238811 and ONR grant N00014-12-1-0049.

A Appendix: Outline of the Proof of Theorem 7

1.1 A.1 Pinch Cographic Matroids

Given an even-cut matroid M where the cycles of M correspond to the even cuts of a graft (G, T) we say that (G, T) is a graft representation of M. An even-cut matroid is pinch-cographic if it has a graft representation with at most four terminals. Consider a graft (H, T) with four terminals, i.e. $T=\{t_1,t_2,t_3,t_4\}$. Let G be obtained from H by identifying vertices $t_1$ and $t_2$ and by identifying vertices $t_3$ and $t_4$. Denote by a the vertex of G corresponding to $t_1=t_2$ and by b the vertex of G corresponding to $t_3=t_4$. Let $\varSigma =\delta _H(t_1)\triangle \delta _H(t_3)$. Then $(G,\varSigma )$ is a signed graph with blocking pair a, b. We say that $(G,\varSigma )$ is obtained from (H, T) by folding and that (H, T) is obtained from $(G,\varSigma )$ by unfolding. Pinch-graphic and pinch-cographic matroids are duals [13], page 26.

Proposition 5

Let $(G,\varSigma )$ be a signed graph with a blocking pair and let (H, T) be obtained from $(G,\varSigma )$ by unfolding. Let M be the pinch-graphic matroid with representation $(G,\varSigma )$ and let N be the pinch-cographic matroid with representation (H, T). Then $M^*=N$.

1.2 A.2 Sizes of Equivalence Classes

Recall that a pair of signed graphs are equivalent if they are related by 1-flips, 2-flips, and resigning.

Proposition 6

There exists a constant $c_1$ such that for every non-graphic, pinch-graphic matroid that is almost 4-connected, the number of pairwise equivalent blocking pair representations is at most $c_1 r(M)^3$ where r(M) denotes the rank of M.

A pair of grafts (H, T) and $(H',T')$ are equivalent if they have the same even-cuts and H and $H'$ are related by 1-flips and 2-flips.

Proposition 7

There exists a constant $c_2$ such that for every non-cographic, pinch-cographic matroid that is almost 4-connected, the number of pairwise equivalent graft representations with four terminals is at most $c_2$.

We will require the following observations [7],

Remark 4

If a pair of signed graphs have the same even-cycles and a common odd cycle then they are equivalent. If a pair of grafts have the same even-cuts and a common odd cut then they are equivalent.

1.3 A.3 Counting Representations

Let $M_1,\ldots ,M_k$ be a good sequence and let $i\in \{1,\ldots ,k\}$. For $i\in \{1,\ldots ,k\}$, let f(i) denote the number of blocking pair representations of $M_i$. We will show,

$$\begin{aligned} f(i)\le 8+6c_2 r(M_i)+ c_1 r(M_i^*) r(M_i)^3 \in {\mathcal O}(|EM_i|^4). \end{aligned}$$

(1)

Proceed by induction on $k-i$. If $k-i=0$, i.e. $M_i=M_k$ then $M_k$ is minimally non-graphic and $f(k)\le 8$. Otherwise $M_{i+1}$ = $M_i$ $\setminus$e or $M_{i+1}=M_i/e$.

Consider first the case where $M_{i+1}=M_i\setminus e$. By induction, (1) holds for $i+1$, i.e. $f(i+1)\le 8+6c_2 r(M_{i+1})+ c_1 r(M_{i+1}^*) r(M_{i+1})^3$. Since $r(M_i^*)=r(M_{i+1}^*)+1$ to prove that (1) holds for $M_i$ we will show $f(i)\le f(i+1)+c_1 r(M_i)^3$. Every blocking pair representation of $M_i$ extends some blocking pair representation of $M_{i+1}$. If each of these representations of $M_{i+1}$ extends to at most one representation of $M_i$ then $f(i)\le f(i+1)$. We prove that if a blocking pair representation of $M_{i+1}$ extends to more than one blocking pair representation of $M_i$ then in each of these representations, e is an odd loop. By Remark 4 each of these representations are pairwise equivalent. By Proposition 6 there are at most $c_1 r(M_i)^3$ of these representations. Thus $f(i)\le f(i+1)+ c_1 r(M_i)^3$ as required.

Consider now the case where $M_{i+1}=M_i/e$. By induction, (1) holds for $i+1$ and since $r(M_i)=r(M_{i+1})+1$ to prove that (1) holds for $M_i$ it suffices to show $f(i)\le f(i+1)+6c_2$. We only consider an example here where we have two distinct blocking pair representations $(G_1,\varSigma )$ and $(G_2,\varSigma )$ of $M_i$ where $(G,\varSigma ):=(G_1,\varSigma )/e=(G_2,\varSigma )/e$ is a representation of $M_{i+1}$ with $\varSigma \subseteq \delta _G(a)\cup \delta _G(b)$ and $a,b\in VG$, and where $G_1$ is obtained from G by splitting vertex a into two vertices, say $a',a''$ so that all edges in $\delta _G(a)\cap \varSigma$ are incident to $a'$ and no edge of $\delta _G(a)\cap \varSigma$ is incident to $a''$ and by joining $a',a''$ by e and where $G_2$ is obtained from G by applying the same construction but now to vertex b. $(G,\varSigma )$ is illustrated in Fig. 4(i) and $(G_1,\varSigma )$ and $(G_2,\varSigma )$ in Fig. 4(ii). We say that $(G_i,\varSigma )$ is obtained from $(G,\varSigma )$ by splitting a signature. How many representations of $M_i$ are obtained in that way? For each representation obtained by splitting a signature, let (H, T) be the graft obtained by unfolding that representation. See Fig. 4(iii) for an illustration where square vertices correspond to terminals. Observe that e is an odd cut of (H, T). By Remark 4, each of these grafts are equivalent. By Propositions 5 and 7 there are at most $c_2$ such grafts. Moreover, every representation of $M_i$ obtained by splitting a signature is obtained by folding such a graft. There are ${4 \atopwithdelims ()2}$ ways of folding a graft, hence, at most $6 c_2$ representations obtained by splitting a signature. Hence, $f(i)\le f(i+1)+6 c_2$ in this case. The analysis for the other cases is similar.

Fig. 4.

Non unique extension and unfolding.