Minimally tough series-parallel graphs with toughness at least $1/2$

TL;DR

This paper characterizes minimally t-tough series-parallel graphs for all t ≥ 1/2, revealing that most such graphs are minimally tough for t > 1/2, but not at t=1/2.

Contribution

It provides a complete characterization of minimally t-tough series-parallel graphs for all t ≥ 1/2, filling a gap in graph toughness theory.

Findings

No minimally t-tough series-parallel graphs exist for t > 1.

Most series-parallel graphs with toughness t > 1/2 are minimally t-tough.

Most series-parallel graphs with toughness 1/2 are not minimally 1/2-tough.

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. Series--parallel graphs are graphs with two distinguished vertices called terminals, formed recursively by two simple composition operations, series and parallel joins. They can be used to model series and parallel electric circuits. We characterize the minimally $t$-tough series-parallel graphs for all $t\ge 1/2$. It is clear that there is no minimally $t$-tough series-parallel graph if $t>1$. We show that for $1\ge t >1/2$, most of the series-parallel graphs with toughness $t$ are minimally $t$-tough, but most of the series-parallel graphs with toughness $1/2$ are not minimally $1/2$-tough.