### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 71-80 |

Number of pages | 10 |

Journal | Discrete Applied Mathematics |

Volume | 161 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 2013 |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Discrete Applied Mathematics*,

*161*(1-2), 71-80. https://doi.org/10.1016/j.dam.2012.08.016