A Characterisation of Pfaffian Near Bipartite Graphs

Towards a Characterisation of Pfaffian graphs

A bipartite graph G is known to be Pfaffian if and only if it does not contain an even subdivision H of K3,3 such that G − V H contains a 1-factor. However a general characterisation of Pfaffian graphs in terms of forbidden subgraphs is currently not known. In this paper we describe a possible approach to the derivation of such a characterisation. We also extend the characterisation for biparti...

Towards a characterisation of Pfaffian near bipartite graphs

A Polynomial Time Algorithm for Recognizing Near-Bipartite Pfaffian Graphs

A matching covered graph is a nontrivial connected graph in which every edge is in some perfect matching. A matching covered graph G is near-bipartite if it is nonbipartite and there is a removable double ear R such that G−R is matching covered and bipartite. In 2000, C. Little and I. Fischer characterized Pfaffian near-bipartite graphs in terms of forbidden subgraphs [3]. However, their charac...

Minimally non-Pfaffian graphs

We consider the question of characterizing Pfaffian graphs. We exhibit an infinite family of non-Pfaffian graphs minimal with respect to the matching minor relation. This is in sharp contrast with the bipartite case, as Little [7] proved that every bipartite non-Pfaffian graph contains a matching minor isomorphic to K3,3. We relax the notion of a matching minor and conjecture that there are onl...

Face-Width of Pfaffian Braces and Polyhex Graphs on Surfaces

A graph G with a perfect matching is Pfaffian if it admits an orientation D such that every central cycle C (i.e. C is of even size and G − V (C) has a perfect matching) has an odd number of edges oriented in either direction of the cycle. It is known that the number of perfect matchings of a Pfaffian graph can be computed in polynomial time. In this paper, we show that every embedding of a Pfa...