In this thesis, two classes of binary matroids will be discussed: even-cycle and even-cut matroids, together with problems which are related to their graphical representations. Even-cycle and even-cut matroids can be represented as signed graphs and grafts, respectively. A signed graph is a pair (G,Σ) where G is a graph and Σ is a subset of edges of G.
A cycle C of G is a subset of edges of G such that every vertex of the subgraph of G induced by C has an even degree. We say that C is even in (G,Σ) if ∣C∩Σ∣ is even. A matroid M is an even-cycle matroid if there exists a signed graph (G,Σ) such that circuits of M precisely corresponds to inclusion-wise minimal non-empty even cycles of (G,Σ). A graft is a pair (G,T) where G is a graph and T is a subset of vertices of G such that each component of G contains an even number of vertices in T. Let U be a subset of vertices of G and let D:=deltaG(U) be a cut of G. We say that D is even in (G,T) if ∣U∩T∣ is even. A matroid M is an even-cut matroid if there exists a graft (G,T) such that circuits of M corresponds to inclusion-wise minimal non-empty even cuts of (G,T).\\
This thesis is motivated by the following three fundamental problems for even-cycle and even-cut matroids with their graphical representations.
(a) Isomorphism problem: what is the relationship between two representations?
(b) Bounding the number of representations: how many representations can a matroid have?
(c) Recognition problem: how can we efficiently determine if a given matroid is in the class? And how can we find a representation if one exists?
These questions for even-cycle and even-cut matroids will be answered in this thesis, respectively. For Problem (a), it will be characterized when two 4-connected graphs G1 and G2 have a pair of signatures (Σ1,Σ2) such that (G1,Σ1) and (G2,Σ2) represent the same even-cycle matroids. This also characterize when G1 and G2 have a pair of terminal sets (T1,T2) such that (G1,T1) and (G2,T2) represent the same even-cut matroid.
For Problem (b), we introduce another class of binary matroids, called pinch-graphic matroids, which can generate expo\-nentially many representations even when the matroid is 3-connected. An even-cycle matroid is a pinch-graphic matroid if there exists a signed graph with a blocking pair. A blocking pair of a signed graph is a pair of vertices such that every odd cycles intersects with at least one of them. We prove that there exists a constant c such that if a matroid is even-cycle matroid that is not pinch-graphic, then the number of representations is bounded by c. An analogous result for even-cut matroids that are not duals of pinch-graphic matroids will be also proven. As an application, we construct algorithms to solve Problem (c) for even-cycle, even-cut matroids. The input matroids of these algorithms are binary, and they are given by a (0,1)-matrix over the finite field \gf(2). The time-complexity of these algorithms is polynomial in the size of the input matrix