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The clique-perfectness and clique-coloring of outer-planar graphs

Author

Listed:
  • Zuosong Liang

    (Qufu Normal University)

  • Erfang Shan

    (Shanghai University)

  • Liying Kang

    (Shanghai University)

Abstract

A clique is defined as a complete subgraph maximal under inclusion and having at least two vertices. 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. A k-clique-coloring of a graph G is an assignment of k colors to the vertices of G such that no clique of G is monochromatic. The smallest integer k admitting a k-clique-coloring of G is called clique-coloring number of G and denoted by $$\chi _{C}(G)$$ χ C ( G ) . In this paper, we first find a class of minimal non-clique-perfect graphs and characterize the clique-perfectness of outer-planar graphs. Secondly, we present a linear time algorithm for the optimal clique-coloring of an outer-planar graph G.

Suggested Citation

  • Zuosong Liang & Erfang Shan & Liying Kang, 2019. "The clique-perfectness and clique-coloring of outer-planar graphs," Journal of Combinatorial Optimization, Springer, vol. 38(3), pages 794-807, October.
    • Handle: RePEc:spr:jcomop:v:38:y:2019:i:3:d:10.1007_s10878-019-00412-2
    DOI: 10.1007/s10878-019-00412-2
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    References listed on IDEAS

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    1. Guillermo Durán & Min Lin & Jayme Szwarcfiter, 2002. "On Clique-Transversals and Clique-Independent Sets," Annals of Operations Research, Springer, vol. 116(1), pages 71-77, October.
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