Recurrence among trees with most numerous efficient dominating sets

Abstract

Graphs and Algorithms International audience 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 is exponential. Also the number of maximum n-vertex trees is shown to be bounded below by an increasing exponential function in n.

The number of maximum matchings in a tree

“…On the other hand, lots of natural graph parameters are defined as the number of vertex or edge subsets of a certain kind; we mention, for example, the number of independent vertex subsets [15,18], the number of matchings [7], the number of dominating or efficient dominating sets [2,3] or the number of subtrees [13,21]. Some of them play an important role in applications as well, for instance the number of matchings that is known as Hosoya index in mathematical chemistry [9,12] and is also connected to the monomer-dimer model of statistical physics [10].…”

“…Remark 3.4. The quasi-periodicity of length 7 is somewhat reminiscent of the situation encountered for dominating sets [2,3], even though there are certain differences.…”