Legal Hop Independent Sequences in Graphs

Abstract

Let G be any graph. A sequence L = (w1, w2, ..., wk) of distinct vertices of G is called a legal sequence if NG[wi]\i[−1j=1NG[wj ] ̸= ∅ for each i ∈ {2, ..., k}. A legal sequence L is called a legal hop independent sequence if dG(wi, wj ) ̸= 2 for each i ̸= j. The maximum length of a legalhop independent sequence in G, denoted by αℓh(G), is called the legal hop independence number of G. In this paper, we initiate the study of legal hop independent sequences in a graph. Weinvestigate its relationships with the hop independence and grundy domination parameter of a graph, respectively. In fact, the legal hop independence parameter is at most equal to the grundy domination (resp. hop independence) parameter on any graph G. Moreover, we derive some formulas or bounds of this parameter on some families of graphs, join, and corona of two graphs.