A New Variant of HOP Independence.

Abstract

Let G=(V(G), E(G)) be a simple and undirected graph. Then P \subseteq V(G) is called a 2-distance certified hop independent set of G if P is a hop independent and for every vertex v in P, v has either zero or at least two hop neighbors in V(G) \setminus P. The 2-distance certified hop independence number of G, denoted by \alpha^2_h(G), is the maximum cardinality among all 2-distance certified hop independent sets of G. In this paper, the study on this new variant of hop independence has been initiated and investigated on some special graphs such as path, and on the join of two graphs. Another concept called clique pointwise was also introduced to give nice properties for a 2-distance certified hop independent sets in a join of two graphs. Moreover, some simplified formulas for the parameter on path and join of two graphs such as fan, wheel, and complete bipartite graphs were derived.