Critical Vertices and Edges in $H$-free Graphs

Daniël Paulusma; Christophe Picouleau; Bernard Ries

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arXiv · 1706.09043

Critical Vertices and Edges in $H$-free Graphs

Daniël Paulusma·Christophe Picouleau·Bernard Ries

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Abstract

A vertex or edge in a graph is critical if its deletion reduces the chromatic number of the graph by 1. We consider the problems of deciding whether a graph has a critical vertex or edge, respectively. We give a complexity dichotomy for both problems restricted to $H$-free graphs, that is, graphs with no induced subgraph isomorphic to $H$. Moreover, we show that an edge is critical if and only if its contraction reduces the chromatic number by 1. Hence, we also obtain a complexity dichotomy for the problem of deciding if a graph has an edge whose contraction reduces the chromatic number by 1.

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BibTeX RIS

Daniël Paulusma, Christophe Picouleau, Bernard Ries. 2017-06-27. Critical Vertices and Edges in $H$-free Graphs. https://arxiv.org/abs/1706.09043

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