On the geodetic iteration number of distance-hereditary graphs

Abstract

For a subset S of vertices of a graph G, let IG0(S)=S, IG1(S)={v:v lies on a shortest pathbetween two vertices of S}, and IGk(S)=IG1(IGk−1(S)) for k≥2. If S=IG1(S) then S is said to be a convex set. The convex hull HG(S) of S is the smallest convex set containing S.In this paper we study the geodetic iteration number of a graph G, defined as the smallest integer k such that HG(S)=IGk(S) for every S⊆V(G). The concept of geodetic iteration number is thus related to the interpretation of IGk(S) in the context of graph convexity. Computing the sequence IG1(S),IG2(S),… can be viewed as a special type of graph dynamics, i.e., the process of iteratively performing applications of a well-defined operator that eventually converges to a fixed point (the convex hull HG(S)). Therefore, the geodetic iteration number is a natural graph invariant that tells us the minimum number of iterations to guarantee the convergence of any initial subset, providing a measure of nonconvexity of the convex space defined over the graph.For each positive integer k, we give a forbidden induced subgraph characterization of distance-hereditary graphs with geodetic iteration number at most k. Distance-hereditary graphs play an important role in the study of geodesic convexity, since for such graphs the distances between vertices are preserved in connected induced subgraphs. As a consequence of our results, we describe a polynomial-time algorithm for the computation of gin(G) when G is a distance-hereditary graph.