Edge Contraction and Line Graphs

TL;DR

This paper investigates conditions under which contracting any edge in a graph preserves certain forbidden subgraph properties, leading to characterizations of claw-free and line graphs.

Contribution

It provides necessary and sufficient conditions for edge contraction to maintain $ ext{H}$-freeness, enabling new characterizations of claw-free and line graphs.

Findings

Characterization of graphs where edge contraction preserves $ ext{H}$-freeness.

Necessary and sufficient conditions for such graphs.

Application to claw-free and line graphs.

Abstract

Given a family of graphs $\mathcal{H}$, a graph $G$ is $\mathcal{H}$-free if any subset of $V(G)$ does not induce a subgraph of $G$ that is isomorphic to any graph in $\mathcal{H}$. We present sufficient and necessary conditions for a graph $G$ such that $G/e$ is $\mathcal{H}$-free for any edge $e$ in $E(G)$. Thereafter, we use these conditions to characterize claw-free and line graphs.