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Relationship Among Triangulations, Quadrangulations and Optimal \(1\)-Planar Graphs

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Abstract

In this paper, we examine relationship of graphs on surfaces: triangulations, quadrangulations and optimal \(1\)-planar graphs. For a given quadrangulation \(G\) of a closed surface \(F^2, G\) can be extended to a triangulation by adding a diagonal edge in every face of \(G\). We show that every quadrangulation of \(F^2\) with at least six vertices can be extended to a \(4\)-connected triangulation. Moreover, we show that every \(5\)-connected triangulation of \(F^2\) has a \(3\)-connected spanning quadrangulation subgraph. As corollaries of these results, we show that every optimal \(1\)-planar graph has a \(4\)-connected triangulation subgraph, and that every plane \(5\)-connected triangulation can be extended to an optimal \(1\)-planar graph by adding some edges.

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Authors and Affiliations

  1. Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

    Kenta Noguchi

  2. Department of Mathematics, Niigata University, 2-8050, ikarashi, Nishi-Ku, Niigata, 950-2181, Japan

    Yusuke Suzuki

Authors
  1. Kenta Noguchi
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  2. Yusuke Suzuki
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Correspondence to Kenta Noguchi.

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Noguchi, K., Suzuki, Y. Relationship Among Triangulations, Quadrangulations and Optimal \(1\)-Planar Graphs. Graphs and Combinatorics 31, 1965–1972 (2015). https://doi.org/10.1007/s00373-015-1568-8

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  • DOI: https://doi.org/10.1007/s00373-015-1568-8

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