Geodetic and hull numbers of strong products of graphs

Classic convexity can be extended to graphs in a natural way by considering shortest paths, also called geodesics: a set S of vertices of a graph is convex if it contains all the vertices lying in some geodesic with endpoints in S and the convex hull of a set S of vertices is the minimum convex set containing S. Farber and Jamison [9] characterized the graphs such that every convex set is the convex hull of its simplicial vertices (i.e. vertices such that its neighborhood induces a complete graph). This rebuilding problem can be studied for general graphs from different points of view. Geodetic and hull numbers give how many vertices are needed, at least, to rebuild the vertex set of a graph by using the closed interval and the convex hull operations respectively. This problem has been studied for different graph classes obtained by means of graph operations. For example, in cartesian products [1,5], compositions [6] and joins [7] of graphs. In this work, we develop these topics for strong products of graphs. All the graphs considered are finite, simple and connected graphs. An x− y path of length d(x, y) is called an x − y geodesic. The closed interval I[x, y] consists of all vertices lying in some x − y geodesic of G. For S ⊆ V (G), the geodetic closure I[S] of S is the union of all closed intervals I[u, v] over all pairs u, v ∈ S, i.e. I[S] = ⋃ u,v∈S I[u, v]. A set S of vertices of G is geodetic if I[S] = V (G) and convex if I[S] = S. The convex hull of S ⊆ V (G) is the smallest convex set containing S and is denoted by CH(S) [8]. A set S ⊆ V (G) is said to be a hull set if its convex hull is the whole vertex set V (G). The geodetic number and the hull number of a graph G are respectively the minimum cardinality among all geodetic sets and hull sets [8,10]. We denote them by g(G) and h(G). Certainly, every geodetic set is a hull set, and hence, h(G) ≤ g(G). The geodetic and hull numbers of paths, cycles, complete graphs, trees and many other classes of graphs is well known. The strong product G£H of graphs G and H is the graph with the vertex set V (G)×V (H) = {(g, h) : g ∈ V (G), h ∈ V (H)} in which vertices (g, h) and (g′, h′) are adjacent whenever (1) g = g′ and hh′ ∈ E(H), or (2) h = h′ and gg′ ∈ E(G), or (3) gg′ ∈ E(G) and hh′ ∈ E(H). In this work we first study some relations between the geodetic and hull sets of the strong product G £ H, and the geodetic and hull sets of its factor graphs G and H. We prove that if S1 is geodetic in G1 and S2 is geodetic in H, then S1 × S2 is geodetic in G £ H. The same result holds for hull sets. We have also that if S is a