Discrete Mathematics & Theoretical Computer Science, Vol 10, No 1 (2008)

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Recurrence among trees with most numerous efficient dominating sets

Dorota Bród, Zdzisław Skupień


A dominating set D of vertices in a graph G is called an efficient dominating set if the distance between any two vertices in D is at least three. A tree T of order n is called maximum if T has the largest number of efficient dominating sets among all n-vertex trees. A constructive characterization of all maximum trees is given. Their structure has recurring aspects with period 7. Moreover, the number of efficient dominating sets in maximum n-vertex trees is determined and appears to be exponential. Also the number of maximum n-vertex trees is shown to be bounded below by an increasing exponential function in n.

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