Distance k-Cost Effective Sets in the Corona and Lexicographic Product of Graphs

Abstract

Let G be a connected graph and k ≥ 1 be an integer. The open k-neighborhood Nk G(v) of v ∈ V (G) is the set Nk G(v) = {u ∈ V (G) \ {v}: dG(u, v) ≤ k}. A set S of vertices of G is called distance k-cost effective of G if for every vertex u in S, |Nk G(u) ∩ (V (G) \ S)| − |Nk G(u) ∩ S| ≥ 0. The maximum cardinality of a distance k-cost effective set of G is called the upper distance k-cost effective number of G. In this paper, we characterized the distance k-cost effective sets in the corona and lexicographic product of two graphs. Consequently, the bounds or the exact values of the upper distance k-cost effective numbers of these graphs are obtained.