Edge Contraction and Forbidden Induced Graphs

TL;DR

This paper investigates the conditions under which contracting any edge in a graph preserves the property of being $H$-free, providing characterizations for various classes of graphs.

Contribution

It establishes necessary and sufficient conditions for graphs to maintain $H$-freeness after any edge contraction, and characterizes several specific graph classes.

Findings

Characterization of graphs where $G/e$ is $H$-free for all edges $e$

Conditions for forests, claw-free, and other specific graph classes

Insights into the structure of $H$-free graphs under edge contraction

Abstract

A graph $G$ is $H$-free if any subset of $V(G)$ does not induce a subgraph of $G$ that is isomorphic to $H$. Given a graph $H$, we present sufficient and necessary conditions for a graph $G$ such that $G/e$ is $H$-free for any edge $e$ in $E(G)$. Thereafter, we use these conditions to characterize forests, claw-free, $2K_{2}$-free, $C_{4}$-free, $C_{5}$-free, split, and pseudo-split graphs.