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Plane Elementary Bipartite Graphs with Forcing or Anti-forcing Edges

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Abstract

Let G be a plane elementary bipartite graph with more than one finite face. We first characterize forcing edges and anti-forcing edges of G in terms of handles and forcing faces. We then introduce the concept of a nice pair of faces of G, which allows us to provide efficient algorithms to construct plane minimal elementary bipartite graphs from G. We show that if \(G'\) is a minimal elementary spanning subgraph of G, then all vertices of a forcing face of G are contained in a forcing face of \(G'\), and any forcing face of G is a forcing face of \(G'\) if it is still a face of \(G'\). Furthermore, any forcing edge (resp., anti-forcing edge) of G is still a forcing edge (resp., an anti-forcing edge) of \(G'\) if it is still an edge of \(G'\). We conclude the paper with the co-existence property of forcing edges and anti-forcing edges for those plane minimal elementary bipartite graphs that remain 2-connected when any handle of length one (if exists) is deleted.

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Authors and Affiliations

  1. Department of Mathematics, Penn State University, Beaver Campus, Monaca, PA, 15061, USA

    Zhongyuan Che

  2. Department of Mathematics, Penn State University, Greater Allegheny Campus, McKeesport, PA, 15132, USA

    Zhibo Chen

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  1. Zhongyuan Che
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  2. Zhibo Chen
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Correspondence to Zhongyuan Che.

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Che, Z., Chen, Z. Plane Elementary Bipartite Graphs with Forcing or Anti-forcing Edges. Graphs and Combinatorics 35, 959–971 (2019). https://doi.org/10.1007/s00373-019-02051-0

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  • DOI: https://doi.org/10.1007/s00373-019-02051-0

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