Characterizing the forbidden pairs for graphs to be super-edge-connected

TL;DR

This paper characterizes specific forbidden subgraph sets that ensure graphs are super-edge-connected, focusing on cases where the set size is one or two, thus advancing understanding of graph connectivity properties.

Contribution

It provides a complete characterization of forbidden subgraph sets that guarantee super-edge-connectivity in graphs, for sets of size one and two.

Findings

Identifies all single forbidden subgraphs ensuring super-edge-connectivity.

Characterizes pairs of forbidden subgraphs that guarantee super-edge-connectivity.

Extends previous results by covering more general forbidden subgraph conditions.

Abstract

Let $\mathcal{H}$ be a set of given connected graphs. A graph $G$ is said to be $\mathcal{H}$-free if $G$ contains no $H$ as an induced subgraph for any $H\in \mathcal{H}$. The graph $G$ is super-edge-connected if each minimum edge-cut isolates a vertex in $G$. In this paper, except for some special graphs, we characterize all forbidden subgraph sets $\mathcal{H}$ such that every $\mathcal{H}$-free is super-edge-connected for $|\mathcal{H}|=1$ and $2$.