An O(|E(G)|2) algorithm for recognizing Pfaffian graphs of a type of bipartite graphs - Feng, Xing; Zhang, Lianzhu; Zhang, Mingzu (Journal of combinatorial optimization, 2018)
A graph G = ( V , E ) with even number vertices is called Pfaffian if it has a Pfaffian orientation, namely it admits an orientation such that the number of edges of any M -alternating cycle which have the same direction as the traversal direction is odd for some perfect matching M of the graph G . In this paper, we obtain a necessary and sufficient condition of Pfaffian graphs in a type of bipartite graphs. Then, we design an O ( | E ( G ) | 2 ) algorithm for recognizing Pfaffian graphs in this class and constructs a Pfaffian orientation if the graph is Pfaffian. The results improve and generalize some known results.