Algorithms for vertex-partitioning problems on graphs with fixed clique-width

Many vertex-partitioning problems can be expressed within a general framework introduced by Telle and Proskurowski. They showed that optimization problems in this framework can be solved in polynomial time on classes of graphs with bounded tree-width. In this paper, we consider a very similar framework, in relationship with more general classes of graphs: we propose a polynomial time algorithm on classes of graphs with bounded clique-width for all the optimization problems in our framework. These classes of graphs are more general than the classes of graphs with bounded tree-width in the sense that classes of graphs with bounded tree-width have also bounded clique-width (but not necessarily the inverse).Our framework includes problems such as independent (dominating) set, p-dominating set, induced bounded degree subgraph, induced p-regular subgraph, perfect matching cut, graph k-coloring and graph list-k-coloring with cardinality constraints (fixed k). This paper thus provides a second (distinct) framework within which the optimization problems can be solved in polynomial time on classes of graphs with bounded clique-width, after a first framework (called MS1) due to the work of Courcelle, Makowsky and Rotics (for which they obtained a linear time algorithm).

[1]  Michael R. Fellows,et al.  Fixed-Parameter Tractability and Completeness II: On Completeness for W[1] , 1995, Theor. Comput. Sci..

[2]  Bruno Courcelle,et al.  Monadic Second-Order Evaluations on Tree-Decomposable Graphs , 1993, Theor. Comput. Sci..

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

[4]  Udi Rotics,et al.  Polynomial algorithms for partitioning problems on graphs with fixed clique-width (extended abstract) , 2001, SODA '01.

[5]  B. A. Reed,et al.  Algorithmic Aspects of Tree Width , 2003 .

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

[7]  Hans L. Bodlaender,et al.  A Partial k-Arboretum of Graphs with Bounded Treewidth , 1998, Theor. Comput. Sci..

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

[9]  Bruno Courcelle,et al.  Monadic Second-Order Evaluations on Tree-Decomposable Graphs , 1991, Theor. Comput. Sci..

[10]  Michael R. Fellows,et al.  FIXED-PARAMETER TRACTABILITY AND COMPLETENESS , 2022 .

[11]  G DowneyRod,et al.  Fixed-Parameter Tractability and Completeness I , 1995 .

[12]  Johannes H. Hattingh,et al.  Majority domination in graphs , 1995, Discret. Math..

[13]  Pierre Hansen,et al.  Splitting trees , 1997, Discret. Math..

[14]  B. Courcellea,et al.  On the xed parameter complexity of graph enumeration problems de nable in monadic second-order logic , 2000 .

[15]  D. R. Lick,et al.  Graph theory with applications to algorithms and computer science , 1985 .

[16]  Jan Arne Telle,et al.  Algorithms for Vertex Partitioning Problems on Partial k-Trees , 1997, SIAM J. Discret. Math..

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

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

[19]  Detlef Seese,et al.  Easy Problems for Tree-Decomposable Graphs , 1991, J. Algorithms.

[20]  G DowneyRod,et al.  Fixed-parameter tractability and completeness II , 1995 .

[21]  M. Jacobson,et al.  n-Domination in graphs , 1985 .

[22]  Zsolt Tuza,et al.  Graph colorings with local constraints - a survey , 1997, Discuss. Math. Graph Theory.

[23]  Michael U. Gerber,et al.  Algorithmic approach to the satisfactory graph partitioning problem , 2000, Eur. J. Oper. Res..

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

[25]  G. Chang,et al.  ALGORITHMIC ASPECTS OF MAJORITY DOMINATION , 1997 .

[26]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[27]  Bruno Courcelle,et al.  An algebraic theory of graph reduction , 1993, JACM.