Counting Independent Sets in Cocomparability Graphs

Abstract We show that the number of independent sets in cocomparability graphs can be counted in linear time, as can counting cliques in comparability graphs. By contrast, counting cliques in cocomparability graphs and counting independent sets in comparability graphs are #P-complete. We extend these results to counting maximal cliques and independent sets. We also consider the fixed-parameter versions of counting cliques and independent sets of given size k. Finally, we combine the results to show that both counting cliques and independent sets in permutation graphs are in linear time.

[1]  Yoshio Okamoto,et al.  Counting the number of independent sets in chordal graphs , 2008, J. Discrete Algorithms.

[2]  Chien-Min Chen,et al.  Linear-time algorithms for counting independent sets in bipartite permutation graphs , 2017, Inf. Process. Lett..

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

[4]  Leslie Ann Goldberg,et al.  Approximately counting locally-optimal structures , 2014, J. Comput. Syst. Sci..

[5]  Min-Sheng Lin.  Counting independent sets and maximal independent sets in some subclasses of bipartite graphs , 2018, Discret. Appl. Math..

[6]  Kurt Mehlhorn,et al.  Certifying algorithms for recognizing interval graphs and permutation graphs , 2003, SODA '03.

[7]  Jeremy P. Spinrad,et al.  Linear-time modular decomposition and efficient transitive orientation of comparability graphs , 1994, SODA '94.

[8]  Martin E. Dyer,et al.  The Relative Complexity of Approximate Counting Problems , 2000, Algorithmica.

[9]  Alfred V. Aho,et al.  The Transitive Reduction of a Directed Graph , 1972, SIAM J. Comput..

[10]  Fedor V. Fomin,et al.  A Fixed-Parameter Perspective on #BIS , 2017, IPEC.

[11]  Stephan Olariu,et al.  Asteroidal Triple-Free Graphs , 1993, SIAM J. Discret. Math..

[12]  Salil P. Vadhan,et al.  The Complexity of Counting in Sparse, Regular, and Planar Graphs , 2002, SIAM J. Comput..

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

[14]  Martin E. Dyer,et al.  On the relative complexity of approximate counting problems , 2000, APPROX.

[15]  Salil P. Vadhan.  The Complexity of Counting , 1995 .

[16]  Ekkehard Köhler,et al.  Linear Time LexDFS on Cocomparability Graphs , 2014, SWAT.

[17]  Chien-Min Chen,et al.  Counting independent sets in tree convex bipartite graphs , 2017, Discret. Appl. Math..