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

Abstract

Let G be a connected graph and S ⊆ 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 [ u , v ] 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 ) | x ∈ 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.