Abstract
Let G be a connected graph. A subset S \subseteq V(G) is a strong resolving dominating set of G if S is a dominating set and for every pair of vertices u,v \in V(G), there exists a vertex w \in S such that u \in I_G[v,w] or v \in I_G[u,w]. The smallest cardinality of a strong resolving dominating set of G is called the strong resolving domination number of G. In this paper, we characterize the strong resolving dominating sets in the join and corona of graphs and determine the bounds or exact values of the strong resolving domination number of these graphs.
Highlights
All graphs considered in this study are finite, simple, and undirected connected graphs, that is, without loops and multiple edges
This study aims to define and characterize the strong resolving dominating sets and determine the exact values or bounds in the join and corona of two graphs
A proper subset S of V (K1 + G) is a strong resolving dominating set of K1 + G if and only if S = V (G) or S = V (G) \ Ci∗ or S = V (K1 + G) \ Ci where Ci is a superclique in Gi, for i = 1, 2, . . . , m and Ci∗ is a dominated superclique of Gi
Summary
All graphs considered in this study are finite, simple, and undirected connected graphs, that is, without loops and multiple edges. The domination number of a graph G, denoted by γ(G), is given by γ(G) = min{|S| : S is a dominating set of G}. The smallest cardinality of a strong resolving set of G is called the strong metric dimension of G and is denoted by sdim(G). The smallest cardinality of a strong resolving dominating set of G is called the strong resolving domination number of G and is denoted by γsr(G). A strong resolving dominating set of cardinality γsr(G) is called a γsr-set of G. The same concept was introduced by Harary and Melter [4] but using the terms resolving sets and metric dimension to refer to locating sets and locating number, respectively. This study aims to define and characterize the strong resolving dominating sets and determine the exact values or bounds in the join and corona of two graphs
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