Hop Differentiating Hop Dominating Sets in Graphs.

A subset S of V (G), where G is a simple undirected graph, is hop dominating if for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2 and it is hop differentiating if N²G[u] ∩ S ̸= N²G[v] ∩ S for any two distinct vertices u, v ∈ V (G). A set S ⊆ V (G) is hop differentiating hop dominating if it is both hop differentiating and hop dominating in G. The minimum cardinality of a hop differentiating hop dominating set in G, denoted by γdh(G), is called the hop differentiating hop domination number of G. In this paper, we investigate some properties of this newly defined parameter. In particular, we characterize the hop differentiating hop dominating sets in graphs under some binary operations. [ABSTRACT FROM AUTHOR]