2-Distance Certified Independence in the Join of Two Graphs.

Abstract

Given a simple and undirected graph G=(V(G),E(G)). Then R\subseteq V(G) is called a 2-distance certified independent set of G if R satisfies the following two conditions:(i)For every x\in R, x has either zero or at least two hop neighborhood in V(G)\setminus R; and (ii) R is an independent set of G. The 2-distance certified independence number of G, denoted by \alpha_{cer}^2(G), is the maximum cardinality of a 2-distance certified independent set of G. In this paper, we introduce this new variant of independence and investigate this on the join of two graphs. Moreover, we characterize the 2-distance certified independent sets in the join of two graphs using pointwise independence concept, a newly defined variant, and finally derive the formulas for the parameter of the said graphs.