J2- Hop Domination in Graphs: Properties and Connections with other Parameters

A subset T = {v(1), v(2), center dot center dot center dot, v(m)} of vertices of a graph G is called a J(2)-set if N-G(2)[v(i)] \ N-G(2)[v(j)] not equal circle divide for every i not equal j, where i, j is an element of{1, 2, . . . , m}. A J(2)-set T is called a J(2)-hop dominating in G if for every a is an element of V (G) \ T, there exists b is an element of T such that d(G)(a, b) = 2. The J(2)-hop domination number of G, denoted by gamma(J2h)(G), is the maximum cardinality among all J(2)-hop dominating sets in G. In this paper, we initiate the study on J(2)-hop domination and we establish its properties and connections with other known parameters in graph theory. We show that every maximum hop independent set is a J(2)-hop dominating, hence, this parameter is greater than compare to the hop independence parameter on any graph. Moreover, we derive some lower and upper bounds of the parameter for a generalized graph, join and corona of two graphs, respectively. Finally, we obtain exact values of the parameter for some special graphs and shadow graph using the characterization results that are formulated in this study.