On the Steiner, geodetic and hull numbers of graphs

Given a graph G and a subset W ⊆ V (G), a Steiner W -tree is a tree of minimum order that contains all of W . Let S(W ) denote the set of all vertices in G that lie on some Steiner W -tree; we call S(W ) the Steiner interval of W . If S(W ) = V (G), then we call W a Steiner set of G. The minimum order of a Steiner set of G is called the Steiner number of G. Given two vertices u, v in G, a shortest u− v path in G is called a u− v geodesic. Let I[u, v] denote the set of all vertices in G lying on some u − v geodesic, and let J [u, v] denote the set of all vertices in G lying on some induced u− v path. Given a set S ⊆ V (G), let I[S] = ⋃u,v∈S I[u, v], and let J [S] = ⋃u,v∈S J [u, v]. We call I[S] the geodetic closure of S and J [S] the monophonic closure of S. If I[S] = V (G), then S is called a geodetic set of G. If J [S] = V (G), then S is called a monophonic set of G. The minimum order of a geodetic set in G is named the geodetic number of G. In this paper, we explore the relationships both between Steiner sets and geodetic sets and between Steiner sets and monophonic sets. We thoroughly study the relationship between the Steiner number and the geodetic number, and address the questions: in a graph G when must every Steiner set also be geodetic and when must every Steiner set also be monophonic. In particular, among others we show that every Steiner set in a connected graph G must also be monophonic, and that every Steiner set in a connected interval graph H must be geodetic. Preprint submitted to Elsevier Science 6th September 2004