On minimally 2-connected graphs with generalized connectivity $\kappa _{3}=2$

Abstract

For $S\subseteq G$, let $\kappa (S)$ denote the maximum number r of edge-disjoint trees $T_1, T_2, \ldots , T_r$ in G such that $V(T_i)\cap V(T_j)=S$ for any $i,j\in \{1,2,\ldots ,r\}$ and $i\ne j$. For every $2\le k\le n$, the k-connectivity of G, denoted by $\kappa _k(G)$, is defined as $\kappa _k(G)=\hbox {min}\{\kappa (S)| S\subseteq V(G)\ and\ |S|=k\}$. Clearly, $\kappa _2(G)$ corresponds to the traditional connectivity of G. In this paper, we focus on the structure of minimally 2-connected graphs with $\kappa _{3}=2$. Denote by $\mathcal {H}$ the set of minimally 2-connected graphs with $\kappa _{3}=2$. Let $\mathcal {B}\subseteq \mathcal {H}$ and every graph in $\mathcal {B}$ is either $K_{2,3}$ or the graph obtained by subdividing each edge of a triangle-free 3-connected graph. We obtain that $H\in \mathcal {H}$ if and only if $H\in \mathcal {B}$ or H can be constructed from one or some graphs $H_{1},\ldots ,H_{k}$ in $\mathcal {B}$ ($k\ge 1$) by applying some operations recursively.