The monadic second-order logic of graphs XVI : Canonical graph decompositions

This article establishes that the split decomposition of graphs introduced by Cunnigham, is definable in Monadic Second-Order Logic.This result is actually an instance of a more general result covering canonical graph decompositions like the modular decom- position and the Tutte decomposition of 2-connected graphs into 3-connected components. As an application, we prove that the set of graphs having the same cycle matroid as a given 2-connected graph can be defined from this graph by Monadic Second-Order formulas.

[1]  W. T. Tutte.  Connectivity in Matroids , 1966, Canadian Journal of Mathematics.

[2]  W. T. Tutte.  Connectivity in graphs , 1966 .

[3]  Robert E. Tarjan,et al.  Isomorphism of Planar Graphs , 1972, Complexity of Computer Computations.

[4]  J. Edmonds,et al.  A Combinatorial Decomposition Theory , 1980, Canadian Journal of Mathematics.

[5]  Klaus Truemper,et al.  On Whitney's 2-isomorphism theorem for graphs , 1980, J. Graph Theory.

[6]  Michel Habib,et al.  Partitive hypergraphs , 1981, Discret. Math..

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

[8]  F. Radermacher,et al.  Substitution Decomposition for Discrete Structures and Connections with Combinatorial Optimization , 1984 .

[9]  André Bouchet,et al.  Reducing prime graphs and recognizing circle graphs , 1987, Comb..

[10]  James G. Oxley,et al.  Matroid theory , 1992 .

[11]  Andrzej Ehrenfeucht,et al.  A k-Structure Generalization of the Theory of 2-Structures , 1994, Theor. Comput. Sci..

[12]  Bruno Courcelle,et al.  Monadic Second-Order Definable Graph Transductions: A Survey , 1994, Theor. Comput. Sci..

[13]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs VIII: Orientations , 1995, Ann. Pure Appl. Log..

[14]  Tero Harju,et al.  Structure and organization , 2014 .

[15]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs X: Linear Orderings , 1996, Theor. Comput. Sci..

[16]  Christian Capelle.  Block Decomposition of Inheritance Hierarchies , 1997, WG.

[17]  Bruno Courcelle,et al.  The Expression of Graph Properties and Graph Transformations in Monadic Second-Order Logic , 1997, Handbook of Graph Grammars.

[18]  Serafino Cicerone,et al.  On the Extension of Bipartite to Parity Graphs , 1999, Discret. Appl. Math..

[19]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs XI: Hierarchical Decompositions of Connected Graphs , 1999, Theor. Comput. Sci..

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

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

[22]  Cyril Gavoille,et al.  Distance labeling scheme and split decomposition , 2003, Discret. Math..

[23]  Jeremy P. Spinrad,et al.  Efficient graph representations , 2003, Fields Institute monographs.

[24]  Fabien de Montgolfier,et al.  De'composition Modulaire des Graphes. The'orie, Extensions et Algorithmes , 2003 .

[25]  Petr Hlinený,et al.  On Matroid Properties Definable in the MSO Logic , 2003, MFCS.

[26]  Jeremy P. Spinrad,et al.  A linear algorithm to decompose inheritance graphs into modules , 1995, Algorithmica.

[27]  Michael Benedikt,et al.  Towards a Characterization of Order-Invariant Queries over Tame Structures , 2005, CSL.

[28]  Bruno Courcelle,et al.  The Modular Decomposition of Countable Graphs: Constructions in Monadic Second-Order Logic , 2005, CSL.

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

[30]  Bruno Courcelle,et al.  The monadic second-order logic of graphs XV: On a conjecture by D. Seese , 2006, J. Appl. Log..

[31]  Bruno Courcelle,et al.  Vertex-minors, monadic second-order logic, and a conjecture by Seese , 2007, J. Comb. Theory, Ser. B.

[32]  Bruno Courcelle,et al.  Circle graphs and monadic second-order logic , 2008, J. Appl. Log..

[33]  N. White.  Theory of Matroids , 2008 .