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.

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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.