Critical Vertices and Edges in $H$-free Graphs
Computational Complexity
2017-06-29 v1 Discrete Mathematics
Combinatorics
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 -free graphs, that is, graphs with no induced subgraph isomorphic to . 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.
Keywords
Cite
@article{arxiv.1706.09043,
title = {Critical Vertices and Edges in $H$-free Graphs},
author = {Daniël Paulusma and Christophe Picouleau and Bernard Ries},
journal= {arXiv preprint arXiv:1706.09043},
year = {2017}
}