A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs

An output-polynomial algorithm for the listing of minimal dominating sets in graphs is a challenging open problem and is known to be equivalent to the well-known Transversal problem which asks for an output-polynomial algorithm for listing the set of minimal transversals in hypergraphs. We give a polynomial delay algorithm to list the set of minimal dominating sets in chordal graphs, an important and well-studied graph class where such an algorithm was not known. The algorithm uses a new decomposition method of chordal graphs based on clique trees.

[1]  Pim van 't Hof,et al.  Minimal dominating sets in graph classes: Combinatorial bounds and enumeration , 2013, Theor. Comput. Sci..

[2]  F. Gavril.  The intersection graphs of subtrees in tree are exactly the chordal graphs , 1974 .

[3]  Takeaki Uno,et al.  On the Enumeration and Counting of Minimal Dominating sets in Interval and Permutation Graphs , 2013, ISAAC.

[4]  Georg Gottlob,et al.  Identifying the Minimal Transversals of a Hypergraph and Related Problems , 1995, SIAM J. Comput..

[5]  Michel Habib,et al.  Chordal Graphs and Their Clique Graphs , 1995, WG.

[6]  Takeaki Uno,et al.  Polynomial Delay Algorithm for Listing Minimal Edge Dominating Sets in Graphs , 2014, WADS.

[7]  Benno Schwikowski,et al.  On enumerating all minimal solutions of feedback problems , 2002, Discret. Appl. Math..

[8]  Georg Gottlob,et al.  New Results on Monotone Dualization and Generating Hypergraph Transversals , 2003, SIAM J. Comput..

[9]  David Avis,et al.  Reverse Search for Enumeration , 1996, Discret. Appl. Math..

[10]  Yann Strozecki,et al.  Enumeration complexity and matroid decomposition , 2010 .

[11]  G. Dirac.  On rigid circuit graphs , 1961 .

[12]  Boros Endre,et al.  Generating Weighted Transversals of a Hypergraph , 2000 .

[13]  Lhouari Nourine,et al.  On the Enumeration of Minimal Dominating Sets and Related Notions , 2014, SIAM J. Discret. Math..

[14]  Arnaud Mary.  Enumération des dominaux minimaux d'un graphe , 2013 .

[15]  Vladimir Gurvich,et al.  Generating Cut Conjunctions in Graphs and Related Problems , 2007, Algorithmica.

[16]  Vladimir Gurvich,et al.  On Enumerating Minimal Dicuts and Strongly Connected Subgraphs , 2007, Algorithmica.

[17]  Fedor V. Fomin,et al.  Enumerating Minimal Subset Feedback Vertex Sets , 2011, WADS.

[18]  Heikki Mannila,et al.  Fast Discovery of Association Rules , 1996, Advances in Knowledge Discovery and Data Mining.

[19]  Heikki Mannila,et al.  Verkamo: Fast Discovery of Association Rules , 1996, KDD 1996.

[20]  Robert E. Tarjan,et al.  Enumeration of the Elementary Circuits of a Directed Graph , 1972, SIAM J. Comput..

[21]  Petr A. Golovach,et al.  Minimal dominating sets in interval graphs and trees , 2017, Discret. Appl. Math..

[22]  Fedor V. Fomin,et al.  Enumerating Minimal Subset Feedback Vertex Sets , 2011, Algorithmica.

[23]  K. Ramamurthy.  Coherent Structures and Simple Games , 1990 .

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

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

[26]  Vladimir Gurvich,et al.  Generating Partial and Multiple Transversals of a Hypergraph , 2000, ICALP.

[27]  Leonid Khachiyan,et al.  On the Complexity of Dualization of Monotone Disjunctive Normal Forms , 1996, J. Algorithms.

[28]  Vladimir Gurvich,et al.  On theory of multistep games , 1973 .

[29]  Jeffrey D. Uuman.  Principles of database and knowledge- base systems , 1989 .

[30]  Vladimir Gurvich,et al.  Transversal hypergraphs to perfect matchings in bipartite graphs: Characterization and generation algorithms , 2006, J. Graph Theory.

[31]  Vladimir Gurvich,et al.  Dual-Bounded Generating Problems: Partial and Multiple Transversals of a Hypergraph , 2001, SIAM J. Comput..

[32]  Vladimir Gurvich,et al.  Generating All Vertices of a Polyhedron Is Hard , 2006, SODA '06.

[33]  Bruno Courcelle,et al.  Linear delay enumeration and monadic second-order logic , 2009, Discret. Appl. Math..

[34]  Eugene L. Lawler,et al.  Generating all Maximal Independent Sets: NP-Hardness and Polynomial-Time Algorithms , 1980, SIAM J. Comput..

[35]  Petr A. Golovach,et al.  Enumerating minimal dominating sets in chordal bipartite graphs , 2016, Discret. Appl. Math..

[36]  Lhouari Nourine,et al.  On the Neighbourhood Helly of Some Graph Classes and Applications to the Enumeration of Minimal Dominating Sets , 2012, ISAAC.