### 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) |
---|---|

Pages (from-to) | 71-80 |

Number of pages | 10 |

Journal | Discrete Applied Mathematics |

Volume | 161 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1 2013 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*,

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

}

*Discrete Applied Mathematics*, vol. 161, no. 1-2, pp. 71-80. https://doi.org/10.1016/j.dam.2012.08.016

**Forcing faces in plane bipartite graphs (II).** / Che, Zhongyuan; Chen, Zhibo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Forcing faces in plane bipartite graphs (II)

AU - Che, Zhongyuan

AU - Chen, Zhibo

PY - 2013/1/1

Y1 - 2013/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84869085535&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84869085535&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2012.08.016

DO - 10.1016/j.dam.2012.08.016

M3 - Article

VL - 161

SP - 71

EP - 80

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 1-2

ER -