Forcing faces in plane bipartite graphs (II)

• Published in: Discrete Applied Mathematics 2013-01, Vol.161 (1-2), p.71-80
• Main Authors: Che, Zhongyuan, Chen, Zhibo
• Format: Article
• Language: English
• Subjects: [formula omitted]-transformation graph; Counting; Forcing edge; Forcing face; Graphs; Handle; Hexagons; Matching; Mathematical analysis; Mathematical models; Perfect matching; Plane elementary bipartite graph; Planes
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2012.08.016

The concept of forcing faces of a plane bipartite graph was first introduced in Che and Chen (2008) [3] [Z. Che, Z. Chen, Forcing faces in plane bipartite graphs, Discrete Mathematics 308 (2008) 2427–2439], which is a natural generalization of the concept of forcing hexagons of a hexagonal system introduced in Che and Chen (2006) [2] [Z. Che and Z. Chen, Forcing hexagons in hexagonal systems, MATCH Commun. Math. Comput. Chem. 56 (2006) 649–668]. In this paper, we further extend this concept from finite faces to all faces (including the infinite face) as follows: A face s (finite or infinite) of a 2-connected plane bipartite graph G is called a forcing face if the subgraph G−V(s) obtained by removing all vertices of s together with their incident edges has exactly one perfect matching. For a plane elementary bipartite graph G with more than two vertices, we give three necessary and sufficient conditions for G to have all faces forcing. We also give a new necessary and sufficient condition for a finite face of G to be forcing in terms of bridges in the Z-transformation graph Z(G) of G. Moreover, for the graphs G whose faces are all forcing, we obtain a characterization of forcing edges in G by using the notion of handle, from which a simple counting formula for the number of forcing edges follows.