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

With graphs considered as natural models for many network design problems, edge connectivity $κ'(G)$ and maximum number of edge-disjoint spanning trees $τ(G)$ 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 $\overline{κ'}(G)=\max \{κ'(H): H \mbox{ is a subgraph of } G \}$. Motivated by their applications in network design and by the established inequalities $\overline{κ'}(G)\ge κ'(G) \ge τ(G),$ we present the following in this paper: (i) For each integer $k>0$, a characterization for graphs $G$ with the property that $\overline{κ'}(G) \le k$ but for any edge $e$ not in $G$, $\overline{κ'}(G+e)\ge k+1$. (ii) For any integer $n > 0$, a characterization for graphs $G$ with $|V(G)| = n$ such that $κ'(G) = τ(G)$ with $|E(G)|$ minimized.

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