Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I (S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S = {u, v}, then I (S) = I [u, v] is called the interval between u and v and consists of all vertices that lie on some shortest u-v path in G. The smallest cardinality of a set S of vertices such that boolean OR I-u,I-v is an element of S[u, v] = V(G) is called the geodetic number and is denoted by g(G). The smallest cardinality of a set S of vertices of G such that I (S) = V (G) is called the Steiner geodetic number of G and is denoted by sg(G). We show that for distance-hereditary graphs g(G) <=, sg(G) but that g(G)/sg(G) can be arbitrarily large if G is not distance hereditary. An efficient algorithm for finding the Steiner interval for a set of vertices in a distance-hereditary graph is described and it is shown how contour vertices can be used in developing an efficient algorithm for finding the Steiner geodetic number of a distance-hereditary graph. (c) 2006 Elsevier B.V. All rights reserved.