Let G = (V, E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V − S, there exists u ∈ S such that d(u, v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γ h (G). In this paper we characterize the family of trees and unicyclic graphs for which γ h (G) = γt(G) and γ h (G) = γc(G) where γt(G) and γc(G) are the total domination and connected domination numbers of G respectively. We then present the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph of graph are also discussed.
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Let G = (V, E) be a graph with p vertices and q edges. A subsetThe minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γ h (G). The subdivision graph S(G) of a graph G is a graph obtained by subdividing every edge of G exactly once. In this paper, we obtain an upper bound on hop domination number of subdivision graph of any connected graph G in terms of number of edges q, the maximum degree ∆(G) and domination number γ(G) of G. We also characterize the family of connected graphs attaining this bound.
In this paper we characterize the class of trees, block graphs, cactus graphs and cubic graphs for which the chromatic transversal domination number is equal to two.
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