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Let be a simple graph with vertex set and edge set. Let have at least vertices of degree at least , where and b are positive integers. A function is said to be a signed-edge cover of G if

A set S of vertices of a graph G = (V , E) with no isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domination number t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd t (G) is the minimum number of edges that must be subdivided in order to… (More)

For a positive integer k, a k-rainbow dominating function of a graph G is a function f from the vertex set V (G) to the set of all subsets of the set {1, 2,. .. , k} such that for any vertex v ∈ V (G) with f (v) = ∅ the condition u∈N (v) f (u) = {1, 2,. .. , k} is fulfilled, where N (v) is the neighborhood of v. The 1-rainbow domination is the same as the… (More)