Critical Vertices and Edges in $H$-free Graphs

TL;DR

This paper investigates the computational complexity of identifying critical vertices and edges in $H$-free graphs, providing dichotomies based on the structure of $H$, and establishes a link between critical edges and edge contractions affecting the chromatic number.

Contribution

It offers a complexity classification for detecting critical vertices and edges in $H$-free graphs and relates critical edges to edge contractions that reduce chromatic number.

Findings

Complexity dichotomies for critical vertex and edge detection in $H$-free graphs.

Critical edges are characterized by edge contractions that decrease chromatic number.

Provides a unified framework linking critical edges and graph contractions.

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.