Minimally tough series-parallel graphs with toughness at least $1/2$
Gyula Y. Katona, Humara Khan
TL;DR
The paper provides a complete characterization of minimally $t$-tough series--parallel graphs for all $t\ge 1/2$, showing no such graphs exist for $t>1$ and giving a precise structural description for $1/2\le t\le1$. It develops a SP-tree framework, introduces edge-type notions (jump/leap edges), and uses careful tough-set analysis to identify when minimality holds. The key results show that minimally $t$-tough SP-graphs with $1/2<t\le1$ occur exactly without jump-edges (cycles at $t=1$), while minimally $1/2$-tough SP-graphs are precisely series joins of pearls and edges (necklace- and bracelet-based structures). These findings imply the Generalized Kriesell conjecture holds for $t\ge 1/2$ within this graph class and motivate future work for $t<1/2$ and generalized SP-graphs.
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.