Trees having many minimal dominating sets

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.

[1]  Eugene L. Lawler,et al.  A Note on the Complexity of the Chromatic Number Problem , 1976, Inf. Process. Lett..

[2]  Lhouari Nourine,et al.  Enumeration of Minimal Dominating Sets and Variants , 2011, FCT.

[3]  Pim van 't Hof,et al.  Minimal dominating sets in graph classes: Combinatorial bounds and enumeration , 2012, Theor. Comput. Sci..

[4]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[6]  Fedor V. Fomin,et al.  Exact exponential algorithms , 2013, CACM.

[7]  David Eppstein.  Small Maximal Independent Sets and Faster Exact Graph Coloring , 2001, WADS.

[8]  S. Hedetniemi,et al.  Domination in graphs : advanced topics , 1998 .

[9]  Fabrizio Grandoni,et al.  Combinatorial bounds via measure and conquer: Bounding minimal dominating sets and applications , 2008, TALG.

[10]  Zdzislaw Skupien,et al.  Trees with extremal numbers of dominating sets , 2006, Australas. J Comb..

[11]  N. S. Barnett,et al.  Private communication , 1969 .

[12]  Andreas Björklund,et al.  Inclusion--Exclusion Algorithms for Counting Set Partitions , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[13]  Mikko Koivisto,et al.  An O*(2^n ) Algorithm for Graph Coloring and Other Partitioning Problems via Inclusion--Exclusion , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[14]  J. Moon,et al.  On cliques in graphs , 1965 .

[15]  Peter J. Slater,et al.  Fundamentals of domination in graphs , 1998, Pure and applied mathematics.

[16]  Herbert S. Wilf,et al.  The number of maximal independent sets in a tree , 1986 .