Abstract
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.
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Acknowledgements
We are grateful to the anonymous referees for their careful reading and many helpful suggestions that greatly improved our original manuscript.
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The research is supported by National Natural Science Foundation of China (Grant Nos. 11171279, 11471273).
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Feng, X., Zhang, L. & Zhang, M. An \(O(|E(G)|^2)\) algorithm for recognizing Pfaffian graphs of a type of bipartite graphs. J Comb Optim 35, 740–753 (2018). https://doi.org/10.1007/s10878-017-0207-0
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DOI: https://doi.org/10.1007/s10878-017-0207-0