The geodetic domination number for the product of graphs

A subset S of vertices in a graph G is 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 set 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) (\gamma(G),\gamma_g(G)) of G is the minimum cardinality among all geodetic (dominating‎, ‎geodetic dominating) sets in G‎. ‎In this paper‎, ‎we show that if a triangle free graph G has minimum degree at least ۲ and g(G) = ۲‎, ‎then \gamma _g(G) = \gamma(G)‎. ‎It is shown‎, ‎for every nontrivial connected graph G with \gamma(G) = ۲ and diam(G) > ۳‎, ‎that \gamma_g(G) > g(G)‎. ‎The lower bound for the geodetic domination number of Cartesian product graphs is proved‎. ‎Geodetic domination number of product of cycles (paths) are determined‎. In this work‎, ‎we also determine some bounds and exact values of the geodetic domination number of strong product of graphs‎.

Keywords:

Cartesian product , strong product , geodetic number , Domination Number , geodetic domination number