Forbidden pairs for equality of edge-connectivity and minimum degree

TL;DR

This paper characterizes pairs of forbidden subgraphs that ensure connected graphs avoiding these subgraphs have equal edge-connectivity and minimum degree.

Contribution

It provides a complete characterization of all pairs of graphs whose absence guarantees equality of edge-connectivity and minimum degree in connected graphs.

Findings

Identifies all pairs of forbidden graphs with the specified property.

Provides a structural characterization of such graph pairs.

Advances understanding of graph connectivity and degree relationships.

Abstract

Let $\mathcal{H}$ be a class of given graphs. A graph $G$ is said to be $\mathcal{H}$-free if $G$ contains no induced copies of $H$ for any $H \in \mathcal{H}$. In this article, we characterize all pairs $\{R,S\}$ of graphs such that every connected $\{R,S\}$-free graph has the same edge-connectivity and minimum degree.