Transactions on Combinatorics (Dec 2014)
The geodetic domination number for the product of graphs
Abstract
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)(γ(G),γ_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 2 and g(G) = 2, then γ_g(G) = γ(G). It is shown, for every nontrivial connected graph G with γ(G) = 2 and diam(G) > 3, that γ_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.
