Trees having many minimal dominating sets

Authors:
Marcin Krzywkowski
Affiliations:
Faculty of Electronics, Telecommunications and Informatics, Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland
Venue:
Information Processing Letters
Year:
2013

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Abstract

We disprove a conjecture by Skupien that every tree of order n has at most 2^n^/^2 minimal dominating sets. We construct a family of trees of both parities of the order for which the number of minimal dominating sets exceeds 1.4167^n. We also provide an algorithm for listing all minimal dominating sets of a tree in time O(1.4656^n). This implies that every tree has at most 1.4656^n minimal dominating sets.