Homogeneously orderable graphs and the Steiner tree problem

In this paper we introduce homogeneously orderable graphs which are a common generalization of distance-hereditary graphs, dually chordal graphs and homogeneous graphs. We present a characterization of the new class in terms of a tree structure of the closed neighbourhoods of homogeneous sets in 2-graphs which is closely related to the defining elimination ordering. The local structure of homogeneously orderable graphs implies a simple polynomial time recognition algorithm for these graphs. Finally we give a polynomial time solution for the Steiner tree problem on homogeneously orderable graphs which extends the efficient solutions of that problem on distance-hereditary graphs, dually chordal graphs and homogeneous graphs.