Let k be a positive integer and G be a connected graph. The open k-neighborhoodset 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 a distance k- cost effective if for every vertex u in S, |Nk G(u) ∩ Sc| − |NkG(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 a distance k- cost effective set in the join of two graphs. As direct consequences, the bounds or the exact values of the upper distance k- cost effective numbers are determined.
A dominating set D of a graph G = (V, E) is non-split dominating set if V \ D is connected. The non-split domination number of G is the minimum cardinality of a non-split dominating set in G. Let D be a minimum dominating set in G. If a subset D of V \ D is dominating in G, then D is called an inverse dominating set with respect to D. Furthermore, if V \ D is connected, then D is called an inverse non-split dominating set. The inverse non-split domination number of G is the minimum cardinality of an inverse non-split dominating set in G. In this paper, characterization of non-split dominating sets in the join and corona of two graphs are presented. Furthermore, explicit formulas for determining the non-split and inverse nonsplit domination numbers of these graphs are also determined.
In this paper, we provide an upper bound for the intersection number in the join and corona of graphs. Moreover, we give formulas for the intersection number of Kn ◦G, Pn ◦G, Cn ◦G and Crn.
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