Characterizations of minimal graphs with equal edge connectivity and spanning tree packing number

TL;DR

This paper characterizes minimal graphs where edge connectivity equals the spanning tree packing number, focusing on their properties and minimal edge configurations for given vertex counts.

Contribution

It provides new characterizations of graphs with equal edge connectivity and spanning tree packing number, especially minimal graphs with specific connectivity properties.

Findings

Characterization of graphs with maximum subgraph edge connectivity at most k

Identification of minimal graphs with equal edge connectivity and spanning tree packing number for given vertices

Abstract

With graphs considered as natural models for many network design problems, edge connectivity $\kappa'(G)$ and maximum number of edge-disjoint spanning trees $\tau(G)$ of a graph $G$ have been used as measures for reliability and strength in communication networks modeled as graph $G$ (see \cite{Cunn85, Matula87}, among others). Mader \cite{Mader71} and Matula \cite{Matula72} introduced the maximum subgraph edge connectivity $\overline{\kappa'}(G)=\max \{\kappa'(H): H \mbox{ is a subgraph of } G \}$. Motivated by their applications in network design and by the established inequalities \[ \overline{\kappa'}(G)\ge \kappa'(G) \ge \tau(G), \] we present the following in this paper: (i) For each integer $k>0$, a characterization for graphs $G$ with the property that $\overline{\kappa'}(G) \le k$ but for any edge $e$ not in $G$, $\overline{\kappa'}(G+e)\ge k+1$. (ii) For any integer $n > 0$, a characterization for graphs $G$ with $|V(G)| = n$ such that $\kappa'(G) = \tau(G)$ with $|E(G)|$ minimized.