The problem of computing a $2$-T-connected spanning subgraph with minimum number of edges in directed graphs

Abstract: Let $G=(V,E)$ be a strongly connected graph with $|V|\geq 3$. For $T\subseteq V$, the strongly connected graph $G$ is $2$-T-connected if $G$ is $2$-edge-connected and for each vertex $w$ in $T$, $w$ is not a strong articulation point. This concept generalizes the concept of $2$-vertex connectivity when $T$ contains all the vertices in $G$. This concept also generalizes the concept of $2$-edge connectivity when $|T|=0$. The concept of $2$-T-connectivity was introduced by Durand de Gevigney and Szigeti in $2018$. In this paper, we prove that there is a polynomial-time 4-approximation algorithm for the following problem: given a $2$-T-connected graph $G=(V,E)$, identify a subset $E^ {2T} \subseteq E$ of minimum cardinality such that $(V,E^{2T})$ is $2$-T-connected.