Bounds and Algorithms for Geodetic Hulls

Abstract

The paper is devoted to the study of geodetic convex hulls in graphs from a theoretical and practical perspective. The notion of convexity can be transferred from continuous geometry to discrete graph structures by defining a node subset to be (geodetically) convex if all shortest paths between its members do not leave the subset. The geodetic convex hull of a node set W is the smallest convex superset of W. The hull number of a graph is then defined as the size of the smallest node subset S (called hull set) whose convex hull contains all graph nodes. In contrast to the geometric setting, where the point subset on the boundary of the convex hull can be computed in polynomial time, it is NP-hard to decide whether a graph has a hull set of size at most s∈N. We establish novel theoretical bounds for graph parameters related to convex graph structures, and also design practical algorithms for upper and lower bounding the hull number. We evaluate the quality of our bounds as well as the performance of the proposed algorithms on road networks and wireless sensor networks of varying size.