Hull number: P5-free graphs and reduction rules

In this paper, we study the (geodesic) hull number of graphs. For any two vertices u,v∈V of a connected undirected graph G=(V,E), the closed interval I[u,v] of u and v is the set of vertices that belong to some shortest (u,v)-path. For any S⊆V, let I[S]=⋃u,v∈SI[u,v]. A subset S⊆V is (geodesically) convex if I[S]=S. Given a subset S⊆V the convex hull Ih[S] of S is the smallest convex set that contains S. We say that S is a hull set of G if Ih[S]=V. The size of a minimum hull set of G is the hull number of G, denoted by hn(G).First, we show a polynomial-time algorithm to compute the hull number of any P5-free triangle-free graph. Then, we present four reduction rules based on vertices with the same neighborhood. We use these reduction rules to propose a fixed parameter tractable algorithm to compute the hull number of any graph G, where the parameter can be the size of a vertex cover of G or, more generally, its neighborhood diversity, and we also use these reductions to characterize the hull number of the lexicographic product of any two graphs.