On the geodetic and geodetic domination numbers of a graph

Abstract

A subset S of vertices in a graph G is a called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S . A subset D of vertices in G is called dominating if every vertex not in D has at least one neighbor in D . A geodetic dominating set S is both a geodetic and a dominating set. The geodetic (domination, geodetic domination) number g ( G ) ( γ ( G ) , γ g ( G ) ) of G is the minimum cardinality among all geodetic (dominating, geodetic dominating) sets in G . In this paper, we study both concepts of geodetic and geodetic dominating sets and derive some upper bounds on the geodetic and the geodetic domination numbers. In particular, we show that if G has minimum degree at least 2 and girth at least 6, then γ g ( G ) = γ ( G ) . We also show that the problem of finding a minimum geodetic dominating set is NP-hard even for chordal or chordal bipartite graphs. Moreover, we present some Nordhaus–Gaddum-type results and study the geodetic and geodetic domination numbers of block graphs.