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Clique-Perfectness of Claw-Free Planar Graphs

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Abstract

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. A graph G is clique-perfect if the sizes of a minimum clique-transversal and a maximum clique-independent set are equal for every induced subgraph of G. An open problem concerning clique-perfect graphs is to find all minimal forbidden induced subgraphs of clique-perfect graphs. In this paper we characterize the clique-perfectness of claw-free planar graphs by forbidden induced subgraphs. Furthermore, we also characterize the clique-perfectness of claw-free graphs with degree at most 4 and graphs with degree at most 3. As an immediate corollary, we show that claw-free cubic graphs are clique-perfect.

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Authors and Affiliations

  1. School of Management, Qufu Normal University, Rizhao, 276800, China

    Zuosong Liang

  2. School of Management, Shanghai University, Shanghai, 200444, China

    Erfang Shan

  3. Mathematics Department, Shanghai University, Shanghai, 200444, China

    Liying Kang

Authors
  1. Zuosong Liang
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  2. Erfang Shan
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  3. Liying Kang
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Corresponding author

Correspondence to Erfang Shan.

Additional information

This research was partially supported by the National Nature Science Foundation of China (Nos. 11571222 and 11471210), the Postdoctoral Foundation of China (No. 2016M592156) and the Nature Science Foundation of Shandong Province, China (No. ZR2014AQ008).

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Liang, Z., Shan, E. & Kang, L. Clique-Perfectness of Claw-Free Planar Graphs. Graphs and Combinatorics 32, 2551–2562 (2016). https://doi.org/10.1007/s00373-016-1726-7

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  • DOI: https://doi.org/10.1007/s00373-016-1726-7

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