Hull and Geodetic Numbers for Some Classes of Oriented Graphs

An oriented graph D is an orientation of a simple graph, i.e. a directed graph whose underlying graph is simple. A directed path from u to v with minimum number of arcs in D is an (u, v)-geodesic, for every u, v ∈ V(D). A set S ⊆ V(D) is (geodesically) convex if, for every u, v ∈ S, all the vertices in each (u, v)-geodesic and in each (v, u)-geodesic are in S. For every S ⊆ V(D) the (convex) hull of S is the smallest convex set containing S and it is denoted by [S]. A hull set of D is a set S ⊆ V(D) whose hull is V(D). The cardinality of a minimum hull set is the hull number of D and it is denoted by $\overrightarrow{hn}\left(D\right)$. A geodetic set of D is a set S ⊆ V(D) such that each vertex of D lies in an (u, v)-geodesic, for some u, v ∈ S. The cardinality of a minimum geodetic set is the geodetic number of D and it is denoted by $\overrightarrow{gn}\left(D\right)$.

In this work, we first present an upper bound for the hull number of oriented split graphs. Then, we turn our attention to the computational complexity of determining such parameters. We first show that computing $\overrightarrow{hn}\left(D\right)$ is NP-hard for partial cubes, a subclass of bipartite graphs, and that computing $\overrightarrow{gn}\left(D\right)$ is also NP-hard for directed acyclic graphs (DAG). Finally, we present a positive result by showing how to compute such parameters in polynomial time when the input graph is an oriented cactus.