On parse trees and Myhill-Nerode-type tools for handling graphs of bounded rank-width

Rank-width is a structural graph measure introduced by Oum and Seymour and aimed at better handling of graphs of bounded clique-width. We propose a formal mathematical framework and tools for easy design of dynamic algorithms running directly on a rank-decomposition of a graph (on contrary to the usual approach which translates a rank-decomposition into a clique-width expression, with a possible exponential jump in the parameter). The main advantage of this framework is a fine control over the runtime dependency on the rank-width parameter. Our new approach is linked to a work of Courcelle and Kante [7] who first proposed algebraic expressions with a so-called bilinear graph product as a better way of handling rank-decompositions, and to a parallel recent research of Bui-Xuan, Telle and Vatshelle.

[1]  Petr Hliněný,et al.  Automata formalization for graphs of bounded rank-width , 2008 .

[2]  Petr Hliněný,et al.  Finding branch-decomposition and rank-decomposition , 2008 .

[3]  Stefan Arnborg,et al.  Problems Easy for Tree-Decomposable Graphs (Extended Abstract) , 1988, ICALP.

[4]  Johann A. Makowsky,et al.  Computing Graph Polynomials on Graphs of Bounded Clique-Width , 2006, WG.

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

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

[7]  Martin Grohe,et al.  The complexity of first-order and monadic second-order logic revisited , 2004, Ann. Pure Appl. Log..

[8]  Jan Arne Telle,et al.  H-join decomposable graphs and algorithms with runtime single exponential in rankwidth , 2010, Discret. Appl. Math..

[9]  Petr Hlinený,et al.  Finding Branch-Decompositions and Rank-Decompositions , 2007, SIAM J. Comput..

[10]  Robert Ganian,et al.  Automata approach to graphs of bounded rank-width , 2008, IWOCA.

[11]  Jan Arne Telle,et al.  H-join and algorithms on graphs of bounded rankwidth (SODA submission) a , 2008 .

[12]  Gian-Carlo Rota,et al.  THE NUMBER OF SUBSPACES OF A VECTOR SPACE. , 1969 .

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

[14]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[15]  Sang-il Oum,et al.  Rank-width and vertex-minors , 2005, J. Comb. Theory, Ser. B.

[16]  B. Mohar,et al.  Graph Minors , 2009 .

[17]  Petr Hlinený,et al.  Branch-width, parse trees, and monadic second-order logic for matroids , 2003, J. Comb. Theory, Ser. B.

[18]  Detlef Seese,et al.  Problems Easy for Tree-Decomposable Graphs (Extended Abstract) , 1988, ICALP.

[19]  Bruno Courcelle,et al.  Graph Operations Characterizing Rank-Width and Balanced Graph Expressions , 2007, WG.

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

[21]  Petr A. Golovach,et al.  Clique-width: on the price of generality , 2009, SODA.

[22]  Martin Grohe,et al.  The complexity of first-order and monadic second-order logic revisited , 2002, Proceedings 17th Annual IEEE Symposium on Logic in Computer Science.

[23]  Michaël Rao,et al.  MSOL partitioning problems on graphs of bounded treewidth and clique-width , 2007, Theor. Comput. Sci..

[24]  Hans-Jürgen Bandelt,et al.  Distance-hereditary graphs , 1986, J. Comb. Theory, Ser. B.

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

[26]  Michael R. Fellows,et al.  Finite automata, bounded treewidth, and well-quasiordering , 1991, Graph Structure Theory.

[27]  Robert Ganian,et al.  Better Polynomial Algorithms on Graphs of Bounded Rank-Width , 2009, IWOCA.

[28]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

[29]  Paul D. Seymour,et al.  Graph minors. X. Obstructions to tree-decomposition , 1991, J. Comb. Theory, Ser. B.

[30]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

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

[32]  Bruno Courcelle,et al.  Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width , 2000, Theory of Computing Systems.

[33]  Bruno Courcelle,et al.  Graph operations characterizing rank-width , 2009, Discret. Appl. Math..