Geodetic and Steiner geodetic sets in 3-Steiner distance hereditary graphs

Let G be a connected graph and S subset of V (G). Then the Steiner distance of S, denoted by d(G) (S), is the smallest number of edges in a connected subgraph of G containing S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval for a set S of vertices in a graph, denoted by I (S) is the union of all vertices that belong to some Steiner tree for S. If S = {u, v}, then I (S) is the interval I vertical bar u, v vertical bar between u and v. A connected graph G is 3-Steiner distance hereditary (3-SDH) if, for every connected induced subgraph H of order at least 3 and every set S of three vertices of H, d(H) (S) = d(G) (S). The eccentricity of a vertex v in a connected graph G is defined as e(v) = max{d(v, x) vertical bar x is an element of V (G)}. A vertex v in a graph G is a contour vertex if for every vertex u adjacent with v, e(u) <= e(v). The closure of a set S of vertices, denoted by I [S], is defined to be the union of intervals between pairs of vertices of S taken over all pairs of vertices in S. A set of vertices of a graph G is a geodetic set if its closure is the vertex set of G. The smallest cardinality of a geodetic set of G is called the geodetic number of G and is denoted by g(G). A set S of vertices of a connected graph G is a Steiner geodetic set for G if I (S) = V (G). The smallest cardinality of a Steiner geodetic set of G is called the Steiner geodetic number of G and is denoted by sg(G). We show that the contour vertices of 3-SDH and HHD-free graphs are geodetic sets. For 3-SDH graphs we also show that g(G) <= sg(G). An efficient algorithm for finding Steiner intervals in 3-SDH graphs is developed. (c) 2007 Elsevier B.V. All rights reserved.