On minimally tough chordal graphs

TL;DR

This paper investigates the properties of minimally tough chordal graphs, providing evidence that such graphs do not exist for certain toughness values and proposing a conjecture for all values greater than one.

Contribution

It extends the understanding of minimally tough chordal graphs by conjecturing their non-existence for all toughness greater than one and proves several supporting cases.

Findings

No strongly chordal graphs are minimally t-tough for t>1/2.

No split graphs are minimally t-tough for t>1/2.

No chordal graphs with a universal vertex are minimally t-tough for t>1.

Abstract

Katona and Varga showed that for any rational number $t \in (1/2,1]$, no chordal graph is minimally $t$-tough, while Katona and Khan characterized all minimally $t$-tough, chordal graphs with $t \le 1/2$. We conjecture that no chordal graph is minimally $t$-tough for any $t>1$ and prove several results supporting the conjecture. In particular, we show that for any $t>1/2$, no strongly chordal graph is minimally $t$-tough%, no split graph is minimally $t$-tough, and no chordal graph with a universal vertex is minimally $t$-tough.