Efficient enumeration of dominating sets for sparse graphs

Abstract A dominating set D of a graph G is a set of vertices such that any vertex in G is in D or its neighbor is in D . Enumeration of minimal dominating sets in a graph is one of the central problems in enumeration study since enumeration of minimal dominating sets corresponds to the enumeration of minimal hypergraph transversals. The output-polynomial time enumeration of minimal hypergraph transversals is an interesting open problem. On the other hand, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a k -degenerate graph in O k time per solution using O n + m space, where n and m are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, H -minor free graphs with some fixed H . The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine.

[1]  Kazuhisa Makino,et al.  New Algorithms for Enumerating All Maximal Cliques , 2004, SWAT.

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

[3]  D. R. Lick,et al.  k-Degenerate Graphs , 1970, Canadian Journal of Mathematics.

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

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

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

[7]  Takeaki Uno,et al.  An Optimal Algorithm for Scanning All Spanning Trees of Undirected Graphs , 1997, SIAM J. Comput..

[8]  Roberto Grossi,et al.  Sublinear-Space Bounded-Delay Enumeration for Massive Network Analytics: Maximal Cliques , 2016, ICALP.

[9]  Leland L. Beck,et al.  Smallest-last ordering and clustering and graph coloring algorithms , 1983, JACM.

[10]  Petr A. Golovach,et al.  An Incremental Polynomial Time Algorithm to Enumerate All Minimal Edge Dominating Sets , 2014, Algorithmica.

[11]  Roberto Grossi,et al.  Optimal Listing of Cycles and st-Paths in Undirected Graphs , 2012, SODA.

[12]  Marthe Bonamy,et al.  Enumerating Minimal Dominating Sets in Kt-free Graphs and Variants , 2020, ACM Trans. Algorithms.

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

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

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

[16]  Vladimir Gurvich,et al.  Generating Maximal Independent Sets for Hypergraphs with Bounded Edge-Intersections , 2004, LATIN.

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

[18]  Jaroslav Nesetril,et al.  Sparsity - Graphs, Structures, and Algorithms , 2012, Algorithms and combinatorics.

[19]  Michael A. Henning,et al.  RAINBOW DOMINATION IN GRAPHS , 2008 .

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

[21]  Andrew Thomason,et al.  The Extremal Function for Complete Minors , 2001, J. Comb. Theory B.

[22]  Marthe Bonamy,et al.  Enumerating minimal dominating sets in triangle-free graphs , 2019, STACS.

[23]  Csilla Bujtás,et al.  Tropical Dominating Sets in Vertex-Coloured Graphs , 2015, WALCOM.

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

[25]  Hiroki Arimura,et al.  Efficient Enumeration of Induced Subtrees in a K-Degenerate Graph , 2014, ISAAC.

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

[27]  C. R. Subramanian,et al.  Girth and treewidth , 2005, J. Comb. Theory, Ser. B.

[28]  Heribert Vollmer,et al.  On the Complexity of Hard Enumeration Problems , 2017, LATA.