A note on 2-vertex-connected orientations

TL;DR

This paper explores extensions of Thomassen's theorem on 2-vertex-connected orientations, proving NP-hardness for mixed graphs and characterizing graphs with a new form of connectivity called 2T-connected orientations.

Contribution

It establishes the NP-hardness of deciding 2-vertex-connected orientations in mixed graphs and provides a characterization of graphs with 2T-connected orientations.

Findings

Deciding 2-vertex-connected orientation in mixed graphs is NP-hard.

Provides a characterization of graphs admitting 2T-connected orientations.

Extends Thomassen's theorem to new connectivity concepts.

Abstract

We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is NP-hard. This answers a question of Bang-Jensen, Huang and Zhu. For the second part, we call a directed graph $D=(V,A)$ $2T$-connected for some $T \subseteq V$ if $D$ is 2-arc-connected and $D-v$ is strongly connected for all $v \in T$. We deduce a characterization of the graphs admitting a $2T$-connected orientation from the theorem of Thomassen.