Minimal toughness in special graph classes

TL;DR

This paper studies the properties of minimally $t$-tough graphs within specific graph classes, focusing on their minimum degree and how to recognize them, to understand their structural characteristics.

Contribution

It introduces the analysis of minimally $t$-tough graphs in chordal, split, claw-free, and $2K_2$-free graph classes, highlighting their minimum degree and recognition challenges.

Findings

Characterization of minimum degree in these graph classes

Recognition algorithms for minimally $t$-tough graphs

Structural properties specific to each class

Abstract

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest $t$ for which the graph is $t$-tough, whereby the toughness of complete graphs is defined as infinity. A graph is minimally $t$-tough if the toughness of the graph is $t$, and the deletion of any edge from the graph decreases the toughness. In this paper, we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the classes of chordal graphs, split graphs, claw-free graphs, and $2K_2$-free graphs.