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

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