The complexity of deciding whether a graph admits an orientation with fixed weak diameter

• Published in: Discrete mathematics and theoretical computer science Vol. 17 no. 3; no. Graph Theory; pp. 31 - 42
• Main Authors: Bensmail, Julien, Duvignau, Romaric, Kirgizov, Sergey
• Format: Journal Article
• Language: English
• Published: Nancy DMTCS 17.02.2016; Discrete Mathematics & Theoretical Computer Science
• Subjects: [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]; Combinatorics; complexity; Computer Science; Discrete Mathematics; Graphs; Mathematical problems; oriented graph; strong diameter; weak diameter; complexity; weak diameter; strong diameter; oriented graph
• ISSN: 1365-8050; 1462-7264; 1365-8050
• DOI: 10.46298/dmtcs.2161

An oriented graph $\overrightarrow{G}$ is said weak (resp. strong) if, for every pair $\{ u,v \}$ of vertices of $\overrightarrow{G}$, there are directed paths joining $u$ and $v$ in either direction (resp. both directions). In case, for every pair of vertices, some of these directed paths have length at most $k$, we call $\overrightarrow{G}$ $k$-weak (resp. $k$-strong). We consider several problems asking whether an undirected graph $G$ admits orientations satisfying some connectivity and distance properties. As a main result, we show that deciding whether $G$ admits a $k$-weak orientation is NP-complete for every $k \geq 2$. This notably implies the NP-completeness of several problems asking whether $G$ is an extremal graph (in terms of needed colours) for some vertex-colouring problems.