Recognizing near-bipartite Pfaffian graphs in polynomial time

Authors:
Alberto Alexandre Assis Miranda;Cláudio Leonardo Lucchesi
Affiliations:
Institute of Computing, University of Campinas, Caixa Postal 6176, 13083-852 Campinas, SP, Brazil;Institute of Computing, University of Campinas, Caixa Postal 6176, 13083-852 Campinas, SP, Brazil
Venue:
Discrete Applied Mathematics
Year:
2010

Citing 2
Cited 0

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

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Abstract

A matching covered graph is a non-trivial connected graph in which every edge is in some perfect matching. A non-bipartite matching covered graph G is near-bipartite if there are two edges e"1 and e"2 such that G-e"1-e"2 is bipartite and matching covered. In 2000, Fischer and Little characterized Pfaffian near-bipartite graphs in terms of forbidden subgraphs [I. Fischer, C.H.C. Little, A characterization of Pfaffian near bipartite graphs, J. Combin. Theory Ser. B 82 (2001) 175-222.]. However, their characterization does not imply a polynomial time algorithm to recognize near-bipartite Pfaffian graphs. In this article, we give such an algorithm. We define a more general class of matching covered graphs, which we call weakly near-bipartite graphs. This class includes the near-bipartite graphs. We give a polynomial algorithm for recognizing weakly near-bipartite Pfaffian graphs. We also show that Fischer and Little's characterization of near-bipartite Pfaffian graphs extends to this wider class.