Properties of minimally $t$-tough graphs

Abstract

A graph $G$ is minimally $t$-tough if the toughness of $G$ is $t$ and the deletion of any edge from $G$ decreases the toughness. Kriesell conjectured that for every minimally $1$-tough graph the minimum degree $\delta (G)=2$. We show that in every minimally $1$-tough graph $\delta (G)\le \frac{n+2}{3}$. We also prove that every minimally $1$-tough claw-free graph is a cycle. On the other hand, we show that for every $t\in \mathbb{Q}$ any graph can be embedded as an induced subgraph into a minimally $t$-tough graph.

Cited by in corpus (1)

• The complexity of recognizing minimally tough graphs