Enumerating minimal dominating sets in triangle-free graphs

It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we prove that this is the case in triangle-free graphs. This answers a question of Kant\'e et al. Additionally, we show that deciding if a set of vertices of a bipartite graph can be completed into a minimal dominating set is a NP-complete problem.

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

[2]  James C. Tiernan,et al.  An efficient search algorithm to find the elementary circuits of a graph , 1970, CACM.

[3]  Lhouari Nourine,et al.  Minimal Dominating Set Enumeration , 2016, Encyclopedia of Algorithms.

[4]  Petr A. Golovach,et al.  Output-Polynomial Enumeration on Graphs of Bounded (Local) Linear MIM-Width , 2017, Algorithmica.

[5]  Peter Damaschke.  Parameterized enumeration, transversals, and imperfect phylogeny reconstruction , 2006, Theor. Comput. Sci..

[6]  John M. Barnard,et al.  Substructure searching methods: Old and new , 1993, J. Chem. Inf. Comput. Sci..

[7]  Georg Gottlob,et al.  New results on monotone dualization and generating hypergraph transversals , 2002, STOC '02.

[8]  M. P. Marcus.  Derivation of Maximal Compatibles Using Boolean Algebra , 1964, IBM J. Res. Dev..

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

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

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

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

[13]  Petr A. Golovach,et al.  An Incremental Polynomial Time Algorithm to Enumerate All Minimal Edge Dominating Sets , 2013, ICALP.

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

[15]  Kunihiro Wasa,et al.  Enumeration of Enumeration Algorithms , 2016, ArXiv.

[16]  Georg Gottlob,et al.  Hypergraph Transversal Computation and Related Problems in Logic and AI , 2002, JELIA.

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

[18]  Min-Jen Jou,et al.  The number of maximal independent sets in triangle-free quasi-tree graphs , 2016 .

[19]  Stephen H. Unger,et al.  Minimizing the Number of States in Incompletely Specified Sequential Switching Functions , 1959, IRE Trans. Electron. Comput..

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

[21]  Georg Gottlob,et al.  Computational aspects of monotone dualization: A brief survey , 2008, Discret. Appl. Math..

[22]  E. A. Akkoyunlu,et al.  The Enumeration of Maximal Cliques of Large Graphs , 1973, SIAM J. Comput..

[23]  Jesper Makholm Byskov.  Enumerating maximal independent sets with applications to graph colouring , 2004, Oper. Res. Lett..

[24]  Takeaki Uno,et al.  A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs , 2015, WG.

[25]  Philip S. Yu,et al.  Substructure similarity search in graph databases , 2005, SIGMOD '05.

[26]  Andrea Marino.  An Application: Biological Graph Analysis , 2015 .

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

[28]  Joshua A. Grochow,et al.  Network Motif Discovery Using Subgraph Enumeration and Symmetry-Breaking , 2007, RECOMB.

[29]  Zsolt Tuza,et al.  The Number of Maximal Independent Sets in Triangle-Free Graphs , 1993, SIAM J. Discret. Math..