Polynomial-time recognition of clique-width ≤3 graphs

Clique-width is a relatively new parameterization of graphs, philosophically similar to treewidth. Clique-width is more encompassing in the sense that a graph of bounded treewidth is also of bounded clique-width (but not the converse). For graphs of bounded clique-width, given the clique-width decomposition, every optimization, enumeration or evaluation problem that can be defined by a monadic second-order logic formula using quantifiers on vertices, but not on edges, can be solved in polynomial time. This is reminiscent of the situation for graphs of bounded treewidth, where the same statement holds even if quantifiers are also allowed on edges. Thus, graphs of bounded clique-width are a larger class than graphs of bounded treewidth, on which we can resolve fewer, but still many, optimization problems efficiently. One of the major open questions regarding clique-width is whether graphs of clique-width at most k, for fixed k, can be recognized in polynomial time. In this paper, we present the first polynomial-time algorithm (O(n^2m)) to recognize graphs of clique-width at most 3.

[1]  Jens Gustedt,et al.  Efficient and practical modular decomposition , 1997, SODA '97.

[2]  Andreas Brandstädt,et al.  Maximum Weight Stable Set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time , 2002, Inf. Process. Lett..

[3]  Bruno Courcelle,et al.  On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic , 2001, Discret. Appl. Math..

[4]  Öjvind Johansson.  NLC2-Decomposition in Polynomial Time , 2000, Int. J. Found. Comput. Sci..

[5]  Udi Rotics,et al.  On the Clique-Width of Some Perfect Graph Classes , 2000, Int. J. Found. Comput. Sci..

[6]  Sang-il Oum,et al.  Approximating rank-width and clique-width quickly , 2008, ACM Trans. Algorithms.

[7]  Egon Wanke,et al.  Line graphs of bounded clique-width , 2007, Discret. Math..

[8]  Michel Habib,et al.  Simpler Linear-Time Modular Decomposition Via Recursive Factorizing Permutations , 2008, ICALP.

[9]  Udi Rotics,et al.  On the Clique-Width of Perfect Graph Classes , 1999, WG.

[10]  Bruno Courcelle,et al.  Upper bounds to the clique width of graphs , 2000, Discret. Appl. Math..

[11]  André Bouchet.  Transforming trees by successive local complementations , 1988, J. Graph Theory.

[12]  Clark F. Olson,et al.  Parallel Algorithms for Hierarchical Clustering , 1995, Parallel Comput..

[13]  Udi Rotics,et al.  Polynomial Time Recognition of Clique-Width \le \leq 3 Graphs (Extended Abstract) , 2000, Latin American Symposium on Theoretical Informatics.

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

[15]  Egon Wanke,et al.  k-NLC Graphs and Polynomial Algorithms , 1994, Discret. Appl. Math..

[16]  Andreas Brandstädt,et al.  Structure and stability number of chair-, co-P- and gem-free graphs revisited , 2003, Inf. Process. Lett..

[17]  Michel Habib,et al.  A survey of the algorithmic aspects of modular decomposition , 2009, Comput. Sci. Rev..

[18]  Vadim V. Lozin,et al.  On the linear structure and clique-width of bipartite permutation graphs , 2003, Ars Comb..

[19]  Rolf H. Möhring,et al.  A Fast Algorithm for the Decomposition of Graphs and Posets , 1983, Math. Oper. Res..

[20]  Eric Torng,et al.  SRPT optimally utilizes faster machines to minimize flow time , 2004, SODA '04.

[21]  Laurent Viennot,et al.  Partition Refinement Techniques: An Interesting Algorithmic Tool Kit , 1999, Int. J. Found. Comput. Sci..

[22]  Michel Habib,et al.  A New Linear Algorithm for Modular Decomposition , 1994, CAAP.

[23]  Udi Rotics,et al.  Edge dominating set and colorings on graphs with fixed clique-width , 2003, Discret. Appl. Math..

[24]  Bruno Courcelle,et al.  Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width , 1998, WG.

[25]  W. Cunningham.  Decomposition of Directed Graphs , 1982 .

[26]  Egon Wanke,et al.  How to Solve NP-hard Graph Problems on Clique-Width Bounded Graphs in Polynomial Time , 2001, WG.

[27]  Johann A. Makowsky,et al.  On the Clique-Width of Graphs with Few P4's , 1999, Int. J. Found. Comput. Sci..

[28]  Udi Rotics,et al.  Clique-Width is NP-Complete , 2009, SIAM J. Discret. Math..

[29]  Jeremy P. Spinrad,et al.  An O(n²) Algorithm for Undirected Split Decompositon , 1994, J. Algorithms.

[30]  Udi Rotics,et al.  On the Relationship Between Clique-Width and Treewidth , 2001, SIAM J. Comput..

[31]  B. Reed,et al.  Polynomial Time Recognition of Clique-Width ≤ 3 Graphs , 2000 .

[32]  Michaël Rao,et al.  NLC-2 Graph Recognition and Isomorphism , 2007, WG.

[33]  Michael U. Gerber,et al.  Algorithms for vertex-partitioning problems on graphs with fixed clique-width , 2003, Theor. Comput. Sci..

[34]  Andreas Brandstädt,et al.  Gem- And Co-Gem-Free Graphs Have Bounded Clique-Width , 2004, Int. J. Found. Comput. Sci..

[35]  D. West.  Introduction to Graph Theory , 1995 .

[36]  Feodor F. Dragan,et al.  New Graph Classes of Bounded Clique-Width , 2002, Theory of Computing Systems.

[37]  Elias Dahlhaus,et al.  Parallel Algorithms for Hierarchical Clustering and Applications to Split Decomposition and Parity Graph Recognition , 2000, J. Algorithms.

[38]  Bruno Courcelle,et al.  Handle-Rewriting Hypergraph Grammars , 1993, J. Comput. Syst. Sci..

[39]  Paul D. Seymour,et al.  Approximating clique-width and branch-width , 2006, J. Comb. Theory, Ser. B.

[40]  Hans L. Bodlaender,et al.  A Tourist Guide through Treewidth , 1993, Acta Cybern..

[41]  Jeremy P. Spinrad,et al.  Modular decomposition and transitive orientation , 1999, Discret. Math..