On the resolving strong domination number of graphs: a new notion

Abstract

The study of metric dimension of graph G has widely given some results and contribution of graph research of interest, including the domination set theory. The dominating set theory has been quickly growing and there are a lot of natural extension of this study, such as vertex domination, edge domination, total domination, power domination as well as the strong domination. In this study, we initiate to combine the two above concepts, namely metric dimension and strong domination set. Thus we have a resolving strong domination set. We have obtained the resolving strong domination number, denoted by γrst(G), of some graphs.

The Nonsplit Resolving Domination Polynomial of a Graph

“…For graphs like paths P n , cycles C n ,complete graphs K n ,complete bipartite graph K r,s , star graph K 1,n−1 , bi-star graph B r,s and friendship graph F n , for join and corona product of graphs, the nonsplit resolving domination polynomial will be found. Some fundamental results which will be required for many of our arguments in this paper are as follows: ( , ) = − 4 , where , , for ≥ ≥ 1is a bistar graph formed from the two star 1, and 1, by joined the central vertices by an edge [22][23][24][25].…”

On the resolving strong domination number of some wheel related graphs

“…Dafik et.al [10] have introduced a new notion about combination of two concepts between metric dimension and strong dominating set, namely resolving strong domination number. Most of these study research results can be attributed to the strong dominating set of G [17] and on the domination number study in [8,9,19,20], and the study on resolving domination number of graphs can be seen in [2].…”

“…satisfies the definition of γ st (G) and dim(G). Dafik et.al [10] have the result of resolving strong domination number of path graph (P n ), wheel graph (W n ), cycle graph (C n ), fan graph (F n ), friendship graph (F r n ) and complete graph (K n ). To construct the proof of strong domination number, we can use two cases.…”

On resolving efficient domination number of path and comb product of special graph

“…Bange, et al [6] presented that if graph G has an efficient dominating set, at that point the cardinality of any efficient dominating set equals the domination number of G .Therefore, the cardinality of all efficient dominating sets are the same. Several previous results on this topic are studied in [9,8,12,24,25,16].…”

“…Recently, Hakim, et al [20] constructed a new stereotype by combining the theory of resolving set and the theory of efficient dominating set which is stated as the resolving efficient dominating set. Z is called the resolving efficient dominating set of any graph G if it is not only statisfy the characteristic of efficient set and dominating set, but also r(x|Z) ̸ = r(y|Z) ∀ x, y ∈ G. γ re G denotes the resolving efficient domination number which is the minimum cardinality of the resolving efficient dominating set of graph G. Farther, There are also results on resolving strong domination number and resolving perfect domination number in [16,5]. In this paper, all graphs studied are the comb product graph.…”