Discrete Mathematics & Theoretical Computer Science (Jan 2008)

Recurrence among trees with most numerous efficient dominating sets

  • Dorota Bród,
  • Zdzisław Skupień

Journal volume & issue
Vol. 10, no. 1

Abstract

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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.