Three domination number and connectivity in graphs

Abstract

In a graph G, a vertex dominates itself and its neighbors. A subset S of V is called a dominating set in G if every vertex in V is dominated by at least one vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set. A set S ⊆ V is called a double dominating set of a graph G if every vertex in V is dominated by at least two vertices in S. The minimum cardinality of a double dominating set is called double domination number of G. The connectivity γ(G) of a connected graph G is the minimum number of vertices whose removal results in a disconnected or trivial graph. In this paper, introduced the concept of three domination in graphs. and we obtain an upper bound for the sum of the three domination number and connectivity of a graph and characterize the corresponding extremal graphs.