On the Geodetic Covers and Geodetic Bases of the Composition \(G[K_m]\)

Abstract

Given a connected graph \(G\) and two vertices \(u\) and \(v\) in \(G\), \(I_G[u, v]\) denotes the closed interval consisting of \(u\), \(v\) and all vertices lying on some \(u\)–\(v\) geodesic of \(G\). A subset \(S\) of \(V(G)\) is called a geodetic cover of \(G\) if \(I_G[S] = V(G)\), where \(I_G[S] = \cup_{u,v\in S} I_G[u, v]\). A geodetic cover of minimum cardinality is called a geodetic basis. In this paper, we give the geodetic covers and geodetic bases of the composition of a connected graph and a complete graph.