On the hull and interval numbers of oriented graphs (Brief Announcement)

Abstract

In this work, for a given oriented graph D, we study its interval and hull numbers, denoted by ⃗in (D) and ⃗hn (D), respectively, in the oriented geodetic, ⃗P3 and ⃗P*3 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 ⃗hn g(D) ≤m(D)-n(D)+2 and that there is at least one such that ⃗hn g(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 ⃗hn P3(D) when the underlying graph of D is split or cobipartite.Moreover, we provide a meta-theorem by proving that if deciding whether ⃗ing(D) ≤k or ⃗hn g(D) ≤k is NP-hard or W[i]-hard parameterized by k, for some i ϵ Z*+, then the same holds even if the underlying graph of D is bipartite. Next, we prove that deciding whether ⃗hn P3(D) ≤ k or ⃗hn P3* (D) ≤ k is W[2]-hard parameterized by k, even if the underlying graph of D is bipartite; that deciding whether ⃗in P3(D) ≤ k or whether ⃗in P3*(D) ≤ k is NP-complete, and the same for ⃗hn P3*(D) ≤ k even if D has no directed cycles and the underlying graph of D is a chordal bipartite graph; and that deciding whether ⃗in P3(D) ≤ k or whether ⃗in P3*(D) ≤ k is W[2]-hard parameterized by k, even if the underlying graph of D is split.Finally, we also argue that the interval and hull numbers in the ⃗P3 and ⃗P*3 convexities can be computed in polynomial time for directed graphs with underlying graph of bounded tree-width by using Courcelle's theorem.