Hull and geodetic numbers for some classes of oriented graphs

Let D be an orientation of a simple graph. Given u,v∈V(D), directed shortest (u,v)-path is (u,v)-geodesic. S⊆V(D) convex if, for every u,v∈S, the vertices in each (u,v)-geodesic and (v,u)-geodesic are S. For (convex) hull S, denoted by [S], smallest set containing if [S]=V(D). geodetic vertex lies (u,v)-geodesic, some u,v∈S. The cardinality minimum (resp. set) G number number) D, hn⃗(D) gn⃗(D)). We first show tight upper bound on hn⃗(D). k∈Z+∗, we prove that deciding hn⃗(D)≤k NP-complete when oriented partial cube; gn⃗(D)≤k W[2]-hard parameterized k has no (c⋅lnn)-approximation algorithm, unless P = NP, even underlying graph bipartite or split cobipartite. also polynomial-time algorithms to compute gn⃗(D) cactus.