On the hull and interval numbers of oriented graphs

Abstract

In this work, for a given oriented graph $D$, we study its interval and hull numbers, respectively, in the oriented geodetic, P3 and P3* convexities. This last one, we believe to be formally defined and first studied in this paper, although its undirected version is well-known in the literature. Concerning bounds, for a strongly oriented graph D, and the oriented geodetic convexity, we prove that $ohng(D)\le m(D)-n(D)+2$ and that there is at least one such that $ohng(D)=m(D)-n(D)$. We also determine exact values for the hull numbers in these three convexities for tournaments, which imply polynomial-time algorithms to compute them. These results allow us to deduce polynomial-time algorithms to compute $ohnp(D)$ when the underlying graph of $D$ is split or cobipartite. Moreover, we provide a meta-theorem by proving that if deciding whether $oing(D)\le k$ or $ohng(D)\le k$ is NP-hard or W[i]-hard parameterized by $k$, for some $i\in {\mathbb{Z}}_{+}^{\ast }$, then the same holds even if the underlying graph of $D$ is bipartite. Next, we prove that deciding whether $ohnp(D)\le k$ or $ohnps(D)\le k$ is W[2]-hard parameterized by $k$, even if $D$ is acyclic and its underlying graph is bipartite; that deciding whether $ohng(D)\le k$ is W[2]-hard parameterized by $k$, even if $D$ is acyclic; that deciding whether $oinp(D)\le k$ or $oinps(D)\le k$ is NP-complete, even if $D$ has no directed cycles and the underlying graph of $D$ is a chordal bipartite graph; and that deciding whether $oinp(D)\le k$ or $oinps(D)\le k$ is W[2]-hard parameterized by $k$, even if the underlying graph of $D$ is split. Finally, also argue that the interval and hull numbers in the oriented P3 and P3* convexities can be computed in cubic time for graphs of bounded clique-width by using Courcelle's theorem.