Grundy Hop Domination in Graphs

Abstract

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. Let S = (v1, v2, · · · , vk) be a sequence of distinct vertices of G and let Sˆ = {v1, v2, . . . , vk}. Then S is a legal closed hop neighborhood sequence of G if N2 G[vi]\∪i−1j=1N 2 G[vj ] ̸= ∅ for each i ∈ {2, · · · , k}. If, in addition, Sˆ is a hop dominating set of G, then S is called a Grundy hop dominating sequence.
The maximum length of a Grundy hop dominating sequence in a graph G, denoted by γ hgr(G), is called the Grundy hop domination number of G. In this paper, we determine some (extreme) values for the Grundy hop domination number. It is pointed out that the Grundy hop domination number is at least equal to the hop domination. Bounds for the Grundy hop domination numbers
of some graphs resulting from some binary operations of two graphs are also obtained.