A POLYNOMIAL ALGORITHM FOR TESTING WHETHER A GRAPH IS 3-STEINER DISTANCE HEREDITARY

Let G be a connected graph and S subset of or equal to V(G). Then the Steiner distance of S in G, denoted by d(G)(S), is the smallest number of edges in a connected subgraph of G that contains S. A connected graph is k-Steiner distance hereditary, k greater than or equal to 2, if for every S subset of or equal to V(G) such that \S\ = k and every connected induced subgraph H of G containing S, d(H)(S) = d(G)(S). A polynomial algorithm for testing whether a graph is 3-Steiner distance hereditary is developed. In addition, a polynomial algorithm for testing whether an arbitrary graph has a cycle of length exceeding t, for any fixed t, without crossing chords is provided.