A set S of vertices in a connected graph G is a resolving hop dominating set of G if S is a resolving set in G and for every vertex v ∈ V (G) \ S there exists u ∈ S such that dG(u, v) = 2. The smallest cardinality of such a set S is called the resolving hop domination number of G. This paper presents the characterizations of the resolving hop dominating sets in the join, corona and lexicographic product of two graphs and determines the exact values of their corresponding resolving hop domination number.
A vertex w in a connected graph G strongly resolves two distinct vertices u and v in V (G) if v is in any shortest u-w path or if u is in any shortest v-w path. A set W of vertices in G is a strong resolving set G if every two vertices of G are strongly resolved by some vertex of W. A set S subset of V (G) is a strong resolving hop dominating set of G if S is a strong resolving set in G and for every vertex v ∈ V (G) \ S there exists u ∈ S such that dG(u, v) = 2. The smallest cardinality of such a set S is called the strong resolving hop domination number of G. This paper presents the characterization of the strong resolving hop dominating sets in the join, corona and lexicographic product of graphs. Furthermore, this paper determines the exact value or bounds of their corresponding strong resolving hop domination number.
Let G be a connected graph. A set W ⊆ V (G) is a resolving hop dominating set of G if W is a resolving set in G and for every vertex v ∈ V (G) \ W there exists u ∈ W such that dG(u, v) = 2. A set S ⊆ V (G) is a 1-movable resolving hop dominating set of G if S is a resolving hop dominating set of G and for every v ∈ S, either S \ {v} is a resolving hop dominating set of G or there exists a vertex u ∈ ((V (G) \ S) ∩ NG(v)) such that (S \ {v}) ∪ {u} is a resolving hop dominating set of G. The 1-movable resolving hop domination number of G, denoted by γ 1 mRh(G) is the smallest cardinality of a 1-movable resolving hop dominating set of G. This paper presents the characterization of the 1-movable resolving hop dominating sets in the join, corona and lexicographic product of graphs. Furthermore, this paper determines the exact value or bounds of their corresponding 1-movable resolving hop domination number.
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