Given a simple graph .

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Abstract. In this paper, we introduce a new variation of edge Roman domination, called edge Roman star domination. An edge Roman star domination function of a graph

Let G =(V, E) be a graph and k be a fixed positive integer. A distance k-dominating set in a graph G, is a set of vertices D in V such that every vertex in V\D is at distance at most k from some vertex in D. The minimum cardinality distance k-dominating set in G is the distance k-domination number γ k . The distance k-domination problem is to find a γ k in G. This problem generalizes the dominating set problem, a central problem in theoretical computer science and is therefore NP-complete for general graphs. The main result of this paper is a better time bound for distance k-domination on permutation graphs by exploiting the structure of permutation in a direct way.

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