Minimally tough chordal graphs with toughness at most $1/2$

TL;DR

This paper characterizes minimally t-tough chordal graphs for all t ≤ 1/2, providing insights into their structure and extending to interval graphs, which helps understand graph toughness properties.

Contribution

It offers a complete characterization of minimally t-tough chordal graphs for t ≤ 1/2, including a corollary for interval graphs, advancing the understanding of graph toughness.

Findings

Characterization of minimally t-tough chordal graphs for t ≤ 1/2

Extension to minimally t-tough interval graphs for t ≤ 1/2

Structural insights into toughness and graph classes

Abstract

Let $t$ be a positive real number. A graph is called \emph{$t$-tough} if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough. 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. A graph is \emph{chordal} if it does not contain an induced cycle of length at least $4$. We characterize the minimally $t$-tough, chordal graphs for all $t\le 1/2$. As a corollary, a characterization of minimally $t$-tough, interval graphs is obtained for $t\le 1/2$.