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arXiv · 1801.02253

Perfect graphs with polynomially computable kernels

Adèle Pass-Lanneau·Ayumi Igarashi·Frédéric Meunier
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Abstract

In a directed graph, a kernel is a subset of vertices that is both stable and absorbing. Not all digraphs have a kernel, but a theorem due to Boros and Gurvich guarantees the existence of a kernel in every clique-acyclic orientation of a perfect graph. However, an open question is the complexity status of the computation of a kernel in such a digraph. Our main contribution is to prove new polynomiality results for subfamilies of perfect graphs, among which are claw-free perfect graphs and chordal graphs. Our results are based on the design of kernel computation methods with respect to two graph operations: clique-cutset decomposition and augmentation of flat edges. We also prove that deciding the existence of a kernel - and computing it if it exists - is polynomial in every orientation of a chordal or a circular-arc graph, even not clique-acyclic.

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Adèle Pass-Lanneau, Ayumi Igarashi, Frédéric Meunier. 2018-01-07. Perfect graphs with polynomially computable kernels. https://arxiv.org/abs/1801.02253

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