Exponential Time Algorithms for the Minimum Dominating Set Problem on Some Graph Classes

The Minimum Dominating Set problem remains NP-hard when restricted to chordal graphs, circle graphs and c-dense graphs (i.e. |E| ≥cn2 for a constant c, 0<c<1/2). For each of these three graph classes we present an exponential time algorithm solving the Minimum Dominating Set problem. The running times of those algorithms are O(1.4173n) for chordal graphs, O(1.4956n) for circle graphs, and $O(1.2303^{(1+\sqrt{1-2c})n})$ for c-dense graphs

[1]  Fabrizio Grandoni,et al.  A note on the complexity of minimum dominating set , 2006, J. Discrete Algorithms.

[2]  J. Mark Keil.  The Complexity of Domination Problems in Circle Graphs , 1993, Discret. Appl. Math..

[3]  Alan A. Bertossi,et al.  Dominating Sets for Split and Bipartite Graphs , 1984, Inf. Process. Lett..

[4]  Kellogg S. Booth,et al.  Dominating Sets in Chordal Graphs , 1982, SIAM J. Comput..

[5]  Rolf Niedermeier,et al.  Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs , 2002, Algorithmica.

[6]  Gerhard J. Woeginger,et al.  Exact (Exponential) Algorithms for the Dominating Set Problem , 2004, WG.

[7]  Ton Kloks.  Treewidth of Circle Graphs , 1996, Int. J. Found. Comput. Sci..

[8]  Fabrizio Grandoni,et al.  Some New Techniques in Design and Analysis of Exact (Exponential) Algorithms , 2005, Bull. EATCS.

[9]  Ingo Schiermeyer.  Problems remaining NP-complette for sparse or dense graphs , 1995, Discuss. Math. Graph Theory.

[10]  Dimitrios M. Thilikos,et al.  Graphs with Branchwidth at Most Three , 1999, J. Algorithms.

[11]  Gerhard J. Woeginger,et al.  Exact Algorithms for NP-Hard Problems: A Survey , 2001, Combinatorial Optimization.

[12]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[13]  Fabrizio Grandoni,et al.  Measure and Conquer: Domination - A Case Study , 2005, ICALP.

[14]  Fedor V. Fomin,et al.  Reports in Informatics , 2005 .

[15]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[16]  A. Brandstädt,et al.  Graph Classes: A Survey , 1987 .