On Exponential Domination of Cm × Cn
نویسندگان
چکیده
An exponential dominating set of graph G = (V,E) is a subset D ⊆ V such that ∑ w∈D( 1 2 )d(v,w)−1 ≥ 1 for every v ∈ V, where d(v, w) is the distance between vertices v and w. The exponential domination number, γe(G), is the smallest cardinality of an exponential dominating set. Lower and upper bounds for γe(Cm × Cn) are determined and it is shown that limm,n→∞ γe(Cm×Cn) mn ≤ 1 13 . Two connections are also established between exponential domination and distance-2 domination: (a) If D is an exponential dominating set of the infinite grid graph such that no two vertices in D are closer than distance 5, then D is a distance-2 dominating set; and (b) For sufficiently large m and n, every distance-2 dominating set of Cm ×Cn is an exponential dominating set.
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