Outer-convex Domination in the Composition and Cartesian Product of Graphs
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Abstract
Let G be a connected simple graph. A set S of vertices of a graph G is an outer-convex dominating set if every vertex not in S is adjacent to some vertex in S and V (G) \ S is a convex set. In this paper we characterize the outer-convex dominating sets in the composition and Cartesian product of two connected graphs. It is shown that the outer-convex domination number of a composition G[H] of two connected graphs G = Pm (m ≥ 3) and H = Kn (n ≥ 2) is equal to 2 if m = 3 and n(m − 4) + 2 if m > 3. The outer-convex domination number of the Cartesian product G□H of two non-complete connected graphs G and H depends on the outer-convex domination number of G and H.
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