Matching structure and the matching lattice
Journal of Combinatorial Theory Series B
Pfaffian orientations 0-1 permanents, and even cycles in directed graphs
Discrete Applied Mathematics - Combinatorics and complexity
A characterisation of Pfaffian near bipartite graphs
Journal of Combinatorial Theory Series B
Towards a characterisation of Pfaffian near bipartite graphs
Discrete Mathematics - Algebraic and topological methods in graph theory
Journal of Combinatorial Theory Series B
A generalization of Little's Theorem on Pfaffian orientations
Journal of Combinatorial Theory Series B
The Pfaffian property of Cartesian products of graphs
Journal of Combinatorial Optimization
| Hi-index | 0.00 |
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 [C.H.C. Little, A characterization of convertible (0,1)-matrices, J. Combin. Theory Ser. B 18 (1975) 187-208] proved that every bipartite non-Pfaffian graph contains a matching minor isomorphic to K"3","3. We relax the notion of a matching minor and conjecture that there are only finitely many (perhaps as few as two) non-Pfaffian graphs minimal with respect to this notion. We define Pfaffian factor-critical graphs and study them in the second part of the paper. They seem to be of interest as the number of near perfect matchings in a Pfaffian factor-critical graph can be computed in polynomial time. We give a polynomial time recognition algorithm for this class of graphs and characterize non-Pfaffian factor-critical graphs in terms of forbidden central subgraphs.