Properties of minimally $t$-tough graphs

TL;DR

This paper investigates properties of minimally $t$-tough graphs, establishing bounds on minimum degree, characterizing minimally $1$-tough claw-free graphs as cycles, and demonstrating embedding of any graph into minimally $t$-tough graphs.

Contribution

It proves bounds on minimum degree for minimally 1-tough graphs, characterizes minimally 1-tough claw-free graphs, and shows that any graph can be embedded into a minimally $t$-tough graph.

Findings

In every minimally 1-tough graph, the minimum degree is at most (n+2)/3.

Every minimally 1-tough claw-free graph is a cycle.

Any graph can be embedded as an induced subgraph into a minimally $t$-tough graph for any rational $t$.

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.