Computational and structural aspects of the geodetic and the hull numbers of shadow graphs

Abstract

Given a set X ⊆ V ( G ) , let [ X ] G denote the set of all vertices of G lying on some shortest path between two vertices of X . If [ X ] G = X , then X is a convex set of G . The convex hull of X , denoted by 〈 X 〉 G , is the smallest convex set containing X . We say that X is a hull set of G if 〈 X 〉 G = V ( G ) and that X is a geodetic set of G if [ X ] G = V ( G ) . The hull number h ( G ) of G is the minimum cardinality of a hull set of G ; and the geodetic number g ( G ) of G is the minimum cardinality of a geodetic set of G . The shadow graph of G , denoted by S ( G ) , is the graph obtained from G by adding a new vertex v ′ for each vertex v of G and joining v ′ to the neighbors of v in G . In this paper, we present sharp bounds for the geodetic and the hull numbers of shadow graphs. We characterize the classes of graphs for which some of these bounds are attained. We also prove for a fixed integer k , that the problem of deciding whether g ( S ( G ) ) ≤ n − k is NP -complete even if the diameter of G is 2; and that the problem of deciding whether h ( S ( G ) ) ≤ n − 1 belongs to P .