On the Geodetic and Hull Numbers of Shadow Graphs

Abstract

Given two vertices u, v in a graph G, a shortest (u, v)-path in G is called an (u, v)-geodesic. Let \(I_G[u, v]\) denote the set of all vertices in G lying on some (u, v)-geodesic. Given a set \(T\subseteq V(G)\), let \(I_G[T]=\cup _{u,v \in T}I_G[u, v]\). If \(I_G[T]=T\), we call T a convex set. The convex hull, denoted by \(\langle T \rangle _G\), is the smallest convex set containing T. A subset T of vertices of a graph G is a hull set if \(\langle T \rangle _G=V(G)\). Moreover, T is a geodetic if \(I_G[T]=V(G)\). The hull number h(G) of a graph G is the minimum size of a hull set. The geodetic number g(G) of G is the minimum size of a geodetic set. The shadow graph, denoted by S(G), of a graph G is the graph obtained from G by adding a new vertex \(v^{\prime }\) for each vertex v of G and joining \(v'\) to the neighbors of v in G. In this paper, we study the geodetic and hull numbers of shadow graphs. Bounds for the geodetic and hull numbers of shadow graphs are obtained and for several classes exact values are determined. Graphs G for which \(g(S(G))\in \{2, 3\}\) are characterized.