Graph of a Graph
نویسنده
چکیده
Let S be the set of minimal dominating sets of graph G and U, W ⊂ S with U ⋃ W = S and U ⋂ W = ∅. A Smarandachely mediate-(U,W ) dominating graph D m(G) of a graph G is a graph with V (D m(G)) = V ′ = V ⋃ U and two vertices u, v ∈ V ′ are adjacent if they are not adjacent in G or v = D is a minimal dominating set containing u. particularly, if U = S and W = ∅, i.e., a Smarandachely mediate-(S, ∅) dominating graph D m(G) is called the mediate dominating graph Dm(G) of a graph G. In this paper, some necessary and sufficient conditions are given for Dm(G) to be connected, Eulerian, complete graph, tree and cycle respectively. It is also shown that a given graph G is a mediate dominating graph Dm(G) of some graph. One related open problem is explored. Finally, some bounds on domination number of Dm(G) are obtained in terms of vertices and edges of G.
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