Width parameters beyond tree-width and their applications

Computer Journal

Volumn 51, Issue 3, 2008, Pages 326-362

  Hliněný, Petr ^a,b   Oum, Sang Il ^c,g   Seese, Detlef ^d   Gottlob, Georg ^e,f  

^a MASARYK UNIVERSITY   (Czech Republic)

^b VSB TECHNICAL UNIVERSITY OF OSTRAVA   (Czech Republic)

^c GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^d UNIVERSITY OF KARLSRUHE   (Germany)

^e UNIVERSITY OF OXFORD   (United Kingdom)

^f VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

^g UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Branch width; Clique width; Complexity; Graph; Hypertree width; MSO logic; Rank width; Tree width

Indexed keywords

ALGORITHMS; COMBINATORIAL OPTIMIZATION; COMPUTATIONAL COMPLEXITY; GRAPH THEORY; PARAMETERIZATION;

BOUNDED TREEWIDTH; BRANCH-WIDTH; CLIQUE-WIDTH; PARAMETERIZED ALGORITHMS; RANK-WIDTH; TREE-WIDTH;

TREES (MATHEMATICS);

EID: 43249119401     PISSN: 00104620     EISSN: 14602067     Source Type: Journal    
DOI: 10.1093/comjnl/bxm052     Document Type: Article

References (223)

1. Bodlaender, H.L. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These special issues are on Parametrized Complexity.
2. Downey, R.G. and Fellows, M.R. (1999) Parameterized Complexity. Monographs in Computer Science. Springer-Verlag, New York.
3. Flum, J. and Grohe, M. (2006) Parameterized Complexity Theory. Springer.
4. Cormen, T.H., Leiserson, C.E., Rivest, R.L. and Stein, C. Introduction to Algorithms (2nd edn). MIT Press, Cambndge, MA.
5. Diestel, R. (2005) Graph Theory. Graduate Texts in Mathematics (3rd edn), Vol. 173, Spiingei-Verlag, Berlin.
6. Beineke, L.W. and Pippert, R.E. (1968) The enumeration of labelled 2-trees. Notices Amer. Math. Soc., 15, 384.
7. Beineke, L.W. and Pippert, R.E. (1969) The number of labeled κ-dimensional trees. J. Combinatorial Theory, 6, 200-205.
8. Rose, D.J. (1974) On simple characterizations of κ-trees. Discrete Math., 7, 317-322.
9. Arnborg, S. and Proskurowski, A. (1989) Linear time algorithms for NP-hard problems restricted to partial κ-trees. Discrete, Appl. Math., 23, 11-24.
10. Arnborg, S. and Proskurowski, A. (1986) Problems on graphs with bounded decomposability. Proc. of the 17th South-Eastern Conf. on Combinatorics, Graph Theory and Computing, Utilitas Mathematica, Winnipeg. Congressus Numerantium, 53, 167-170.
11. Arnborg, S. and Proskurowski, A. (1986) Characterization and recognition of partial 3-trees. SIAM J. Algebraic Discrete Methods, 7, 305-314.
12. Robertson, N. and Seymour, P.D. (1983) Graph minors. I. Excluding a forest. J. Combin. Theory Ser. B, 35, 39-61.
13. Robertson, N. and Seymour, P.D. (1986) Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms, 7, 309-322.
14. Robertson, N. and Seymour, P.D. (1985) Graph minors - a survey. Surveys in Combinatorics 1985, Glasgow. London Mathematical Society. Lecture Note Series. Vol. 103, pp. 153-171. Cambridge University Press, Cambridge.
15. Prokurowski, A. (1984) Separating subgraphs in k-trees: Cables and caterpillars. Discrete Math., 49, 275-285.
16. Seese, D. (1985) Tree-partite graphs and the complexity of algorithms (extended abstract). Fundamentals of Computation Theory, Cottbus, Lecture Notes in Comput. Sci. Vol. 199, pp. 412-421. Springer, Berlin.
17. Seese, D. (1985) Baum-partite Graphen und die Komplexitaet von Algorithmen. Proc. 30th Int. Wiss. Koll., TH Ilmenau, pp. 101-103.
18. Seese, D. (1986) Tree-partite graphs and the complexity of algorithms (Full version of [16]). Preprint P-Math 08/86, Karl- Weierstrass-Institute for Mathematics, Berlin.
19. Brandstädt, A., Le, V.B. and Spinrad, J.P. (1999) Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
20. Garey, M.R. and Johnson, D.S. (1979) Computers and Intractability: A guide to the Theory of NP-Completeness. W.H. Freeman, and Co., San Francisco, CA.
21. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A. and Protasi, M. (1999) Complexity and Approximation. Springer-Verlag, Berlin.
22. Allauzen, C. and Durand, B. (1996) Appendix A: Tiling problems. In Börger, E., Grädel, E. and Gurevich, Y. (eds), The Classical Decision Problem, pp. 407-420. Springer-Verlag.
23. Papadimitriou, C.H. (1994) Computational complexity. Addison-Wesley Publishing Company, Reading, MA.
24. Berger, R. (1966) The undecidability of the domino problem. Mem. Amer. Math. Soc. No., 66, 1-72.
25. Hanf, W. (1974) Nonrecursive tilings of the plane. I. J. Symbolic Logic, 39, 283-285.
26. van Emde Boas, P. (1983) Dominoes are forever. Technical Report 83-04, University of Amsterdam.
27. Harel, D. (1983) Recurring dominoes: Making the highly undecidable highly understandable. In Foundations of Computation theory, Borgholm. Lecture Notes in Computer Science, Vol. 158, pp. 177-194. Springer, Berlin.
28. Seese, D. and Schlottmann, F. (2002) Large grids and local information flow as a reason for high complexity. In Frizelle, G. and Richards, H. (ed.), Tackling Industrial Complexity: The Ideas that Make a Difference. Proc. of the 2002 Conference of the Manufacturing Complexity Network, pp. 193-207. University of Cambridge.
29. Robertson, N. and Seymour, P. (1991) Graph minors. X. Obstructions to tree-decomposition. J. Combin. Theory Ser. B, 52, 153-190.
30. Geelen, J.F., Gerards, B. and Whittle, G. (2007) Excluding a planar graph from GF(q)-representable matroid's. J. Combin. Theory Ser. B doi:10.1016/j.jctb.2007.02.005.
31. Oum, S. (2005) Rank-width and vertex-minors. J. Combin. Theory Ser. B 95, 79-100.
32. Höfting, F., Lengrauer, T. and Wanke, E. (1992) Processing of hierarchically Defined Graphs and Graph Families. In Data Structures and Efficient Algorithms, Berlin, 1991. Lecture Notes in Computer Science. Vol. 594, pp. 44-69. Sprinapr Rprlin
33. Lengauer, T. (1982) The complexity of compacting hierarchically specified layouts of integrated circuits. 23rd annual symposium on foundations of computer science, Chicago, IEEE, New York, 358-368.
34. Lengauer, T. (1989) Hierarchical planarity testing algorithms. J. Assoc. Comput. Mach., 36, 474-509.
35. Lengauer, T. (1987) Efficient algorithms for finding minimum spanning forests of hierarchically defined graphs. J. Algorithms, 8, 260-284.
36. Lengauer, T. and Wagner, K.W. (1992) The correlation between the complexities of the nonhierarchical and hierarchical versions of graph problems. J. Comput. Syst. Sci., 44, 63-93.
37. Lengauer, T. and Wanke, E. (1988) Efficient solution of connectivity problems on hierarchically defined graphs. SIAM J. Comput., 17 1063-1080.
38. Lengauer, T. and Wieners, C. (1987) Efficient solutions of hierarchical systems of linear equations. Computing, 39, 111-132.
39. Seese, D. (2007) Easy Problems for grid-structured graphs. In Preparata, F.P. and Fang, Q. (ed.), Proc. of the Int. Frontiers in Algorithmics Workshop FAW 2007, 1-5 August 2007, Lanzou, China, Lecture Notes in Compututer Science. Springer, 1-12. To appear.
40. Wagner, K. (1984) The complexity of problems concerning graphs with regularities. Technical Report N/84/52, Friedrich-Schiller-Universität Jena, 1984. Extended abstract appeared in Proceedings of the 11th MFCS. Lecture Notes in Computer Science, Vol. 176, 544-552, Springer.
41. Lengauer, T. (1986) Exploiting hierarchy in VLSI design. In Proc. of the Aegean Workshop on Computing on VLSI Algorithms and Architectures Springer-Verlag New York, Inc., New York, NY, USA, 180-193.
42. Lengauer, T. (1990) Combinatorial algorithms for integrated circuit layout. In Applicable Theory in Computer Science. John Wiley & Sons Ltd., Chichester (With a foreword by Bryan Preas).
43. Courcelle, B. and Engelfriet, J. (1995) A logical characterization of the sets of hypergraphs defined by hyperedge replacement grammars. Math. Syst. Theory, 28, 515-552.
44. Courcelle, B., Engelfriet, J. and Rozenberg, G. (1993) Handle-rewriting hypergraph grammars, J. Comput. Syst. Sci., 46, 218-270.
45. Drewes, F., Kreowski, H.-J. and Habel, A. (1997) Hyperedge replacement graph grammars. Handbook of Graph Grammars and Computing by Graph Transformation, Vol. 1. 95-162, World Scientific Publishers, River Edge, NJ.
46. Glikson, A. and Makowsky, J.A. (2003) NCE graph grammars and clique-width. Graph-Theoretic Concepts in Computer Science, October Lecture Notes in Computer Science. Vol. 2880, pp. 237-248. Springer.
47. Engelfriet, J. (1997) Context-free graph grammars. Handbook of Formal Languages, Vol. 3, pp. 125-213. Springer, Berlin.
48. Kim, C. (1997) A hierarchy of eNCE families of graph languages. Theoret. Comput. Sci., 186, 157-169.
49. Courcelle, B. and Makowsky, J.A. (2000) VR and HR graph grammars: A common algebraic framework compatible with monadic second-order logic (extended abstract). Preprint. Accepted to GRATRA'2000.
50. Courcelle, B. and Makowsky, J.A. (2002) Fusion in relational structures and the verification of monadic second-order properties. Math. Struct. Comput. Sci., 12, 203-235 (Theory and applications of graph transformations).
51. Seese, D. (1996) Linear time computable problems and first-order descriptions. Math. Structures Comput. Sci., 6, 505-526.
52. Libkin, L. (2004) Elements of Finite Model Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin
53. Frick, M. and Grohe, M. (2001) Deciding first-order properties of locally tree-decomposable structures. J. ACM, 48, 1184-1206 (electronic).
54. Frick, M. and Grohe, M. (2004) The complexity of first-order and monadic second-order logic revisited. Ann. Pure Appl. Logic, 130, 3-31.
55. Arnborg, S., Lagergren, J. and Seese, D. (1991) Easy problems for tree-decomposable graphs. J. Algorithms, 12, 308-340.
56. Borie, R.B., Parker, R.G. and Tovey, C.A. (1992) Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica, 7, 555-581.
57. Courcelle, B. (1992) The monadic second-order logic of graphs III. Tree-decompositions, minors and complexity issues. RAIRO Inform. Théor. Appl., 26, 257-286.
58. Courcelle, B. and Mosbah, M. (1993) Monadic second-order evaluation on tree-decomposable graphs. Theoret. Comput. Sci., 109, 49-82.
59. Courcelle, B., Makowsky, J.A. and Rotics, U. (2000) Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33, 125-150.
60. Robertson, N. and Seymour, P.D. (1983) Graph minors. III. Planar tree-width. J. Combin. Theory Ser. B, 36, 49-64.
61. Robertson, N. and Seymour, P. (1995) Graph minors. XIII. The disjoint paths problem. J. Combin. Theory Ser. B, 63, 65-110.
62. Robertson, N. and Seymour, P. (2004) Graph minors. XX. Wagner's conjecture. J. Combin. Theory Ser. B., 92. 325-357.
63. Lovász, L. (2006) Graph minor theory. Bull. Amer. Math. Soc. (N.S.), 43, 75-86 (electronic).
64. Fagin, R. (1974) Generalized first-order spectra and polynomial-time recognizable sets.Complexity of Computation. Proc. SIAM-AMS Sympos. Appl. Math., New York. Amer. Math. Soc., Providence, RI, Vol. VII pp. 43-73.
65. Immerman, N. (1998) Descriptive Complexity. Graduate Texts in Computer Science. Springer-Verlag, New York.
66. Ebbinghaus, H.-D., Flum, J. and Thomas, W. (1994) Mathematical Logic (2nd edn). Undergraduate Texts in Mathematics, Springer, New York. (Translated from the German by Margit Meßmer).
67. Barwise, J. (1977) An introduction to first-order logic. Handbook of Mathematical Logic, pp. 5-46, North-Holland Publishing Company.
68. Immerman, N. (1987) Languages that capture complexity classes. SIAM J. Comput., 16, 760-778.
69. Monien, B. (1986) The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete. SIAM J. Algebraic Discrete Methods 7, 505-512.
70. Ebbinghaus, H.-D. and Flum, J. (1999) Finite model theory. Perspectives in Mathematical Logic (2nd ed). Springer-Verlag, Berlin.
71. Gurevich, Y. (1985) Monadic Second-Order Theories. Model-theoretic Logics, pp. 479-506, Perspect. Math. Logic. Springer, New York.
72. Courcelle, B. (1990) The monadic second-order logic of graphs IV: definability properties of equational graphs. Ann. Pure Appl. Logic 49, 193-255.
73. Courcelle, B. (1994) The monadic second order logic of graphs VI: On several representations of graphs by relational structures. Discr. Appl. Math., 54, 117-149.
74. Arnborg, S., Proskurowski, A. and Seese, D. (1991) Monadic second order logic, tree automata and forbidden minors. Computer science logic (Heidelberg, 1990), Lecture Notes in
75. Courcelle, B. (1997) The expression of graph properties and graph transformations in monadic second-order logic. Handbook of Graph Grammars and Computing by Graph Transformation, Vol. 1, pp. 313-400. World Science Publishing, River Edge, NJ.
76. Courcelle, B., Makowsky, J.A. and Rotics, U. (2001) On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discrete Appl. Math., 108, 23-52.
77. Rabin, M.O. (1965) A simple method for undecidability proofs and some applications. Logic, Methodology and Philos. Sci., Proc. 1964 Internat. Congr. pp. 58-68, North-Holland, Amsterdam.
78. Arnborg, S., Lagergren, J. and Seese, D. (1988) Problems easy for tree-decomposable graphs (extended abstract). Automata, Languages and Programming (Tampere, 1988), Lecture Notes in Computer Science. Vol. 317, pp. 38-51. Springer, Berlin.
79. Seese, D. (1992) Interpretability and tree automata: A simple way to solve algorithmic problems on graphs closely related to trees. Tree Automata and Languages, Le Touquet, 1990, Stud. Comput. Sci. Artificial Intelligence. Vol. 10, pp, 83-114. North-Holland, Amsterdam.
80. Rabin, M.O. (1977) Decidable theories. In Barwise, J. (ed.), Handbook of Mathematical Logic, 595-629. North-Holland Publishing Co., Amsterdam.
81. Compton, K.J. and Henson, C.W. (1987) A Uniform Method for Proving Lower Bounds on the Computational Complexity of Logical Theories. Technical Report CRL-TR-04-87, Computing Research Laboratory, University of Michigan.
82. Courcelle, B. (1994) Monadic second-order definable graph transductions: a survey. Theoret. Comput. Sci., 126, 53-75.
83. Thatcher, J.W. and Wright, J.B. (1968) Generalized finite automata theory with an application to a decision problem of second-order logic. Math. Syst. Theory, 2, 57-81.
84. Takamizawa, K., Nishezeki, T. and Saito, N. (1982) Linear-time computability of combinatorial problems on series-parallel graphs. J. Assoc. Comput. Mach., 29, 623-641.
85. Scheffler, P. and Seese, D. (1989) A combinatorial and logical approach to linear-time computability. In EUROCAL '87: Proc. of the European Conference on Computer Algebra, Lecture Notes in Computer Science. Vol. 378, pp. 379-380. Springer, London.
86. Wimer, T.V. (1987) Linear algorithms on k-terminal graphs. PhD Thesis, Department of Computer Science, Clemson University, SC.
87. Bern, M.W., Lawler, E.L. and Wong, A.L. (1987) Linear-time computation of optimal subgraphs of decomposable graphs. J. Algorithms, 8 216-235.
88. Bodlaender, H.L. (1988) Dynamic programming on graphs with bounded treewidth. Automata, Languages and Programming, Tampere, Lecture Notes in Computer Science. Vol. 317, pp. 105-118, Springer, Berlin.
89. Fellows, M.R. and Langston, M.A. (1989) An analogue of the Myhill-Nerode theorem and its use in computing finite basis characterizations. Proc. 30th Symp. Foundations of Comp. Sci., pp. 520-525. IEEE Press
90. Hohberg, W. and Reischuk, R. (1989) Decompositions of graphs - a uniform approach for the design of fast sequential and parallel algorithms on graphs. Technical report, TH Darmstadt.
91. Hohberg, W and Reischuk R. (1990) A framework to design algorithms for optimization problems on graphs. Preprint, TH Darmstadt.
92. Bodlaender, H.L. and Thilikos, D.M. (1999) Graphs with branchwidth at most three. J. Algorithms, 32, 167-194.
93. Dharmatilake, J.S. (1994) Binary matroids of branch-width 3. PhD Thesis. Ohio State University.
94. Riggins, K.D. (2001) On characterizing graphs with branchwidth at most four. Mater's Thesis, Rice University.
95. Paul, C., Proskurowski, A. and Telle, J.A. (2006) Generation of graphs with bounded branchwidth. Graph-theoretic Concepts in Computer Science, Bergen, Norway, Lecture Notes in Computer Science, Vol. 4271, pp. 205-216. Springer, Berlin/Heidelberg.
96. Seymour, P.D. and Thomas, R. (1994) Call routing and the ratcatcher. Combinatorica, 14, 217-241.
97. Bodlaender, H.L. and Thilikos, D.M. (1997) Constructive linear time algorithms for branchwidth. Automata, Languages and Programming, Bologna, Lecture Notes in Computer Science, Vol. 1256, pp. 627-637. Springer, Berlin.
98. Hicks, I.V. (2005) Planar branch decompositions. I. The ratcatcher. INFORMS J. Comput., 17, 402-412.
99. Hicks, I.V. (2005) Planar branch decomposition. II. The cycle method. INFORMS J. Comput., 17, 413-421.
100. Formin, F.V. and Thilikos, D.M. (2003) Dominating sets in planar graphs: branch-width and exponential speed-up. Proc. of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, MD, pp. 168-177, ACM, New York.
101. Dorn, F., Penninkx, E., Bodlaender, H.L. and Fomin, F.V. (2005) Efficient exact algorithms on planar graphs: Exploiting sphere cut branch decompositions. Proc. of the 13th Annual European Symposium on Algorithms (ESA 2005), Lecture Notes in Computer Science, Vol. 3669, pp. 95-106.
102. Noble, S.D. (1998). Evaluating the Tutte polynomial for graphs of bounded tree-width. Combin. Probab. Comput. 7, 307-321.
103. Hliněný, P. (2006) The Tutte polynomial for matroids of bounded branch-width. Combin. Probab. Comput., 15, 397-409.
104. Valiant, L.G. (1979) The complexity of enumeration and reliability problems. SIAM J. Comput., 8, 410-421.
105. Oxley, J.G. (1992) Matroid theory. Oxford University Press, New York.
106. Hicks, I.V. and McMurray, N.B., Jr. (2007) The branch-width of graphs and their cycle matroids. J. Combin. Theory, Ser, B., 97, 681-692.
107. Mazoit, F. and Thomassé, S. (2005) Branchwidth of graphic matroids, Preprint.
108. Hliněný, P. (2002) On the excluded minors for matroids of branch-width three. Electron. J. Combin.,9, Research Paper 32, 13 pp. (electronic).
109. Hliněný, P. (2004) Using a computer in matroid theory research. Acta Univ. M. Belii Ser. Math., 11, 27-44.
110. Geelen, J.F., Gerards, A.M.H. and Whittle, G. (2007) Branch-width and well-quasi-ordering and matroid and graphs. J. Combin. Theory Ser. B 84, 270-290.
111. Geelen, J.F., Gerards, A.M.H., Robertson, N. and Whittle, G. (2003) On the excluded minors for the matroids of branch-width. k. J. Combin. Theory Ser. B, 88, 261-265.
112. Oum, S. and Seymour, P. (2007) Testing branch-width. J. Combin. Theory Ser. B, 97, 385-393.
113. Hliněný, P. (2005) A parametrized algorithm for matroid branch-width. SIAM J. Comput., 35, 259-277 (loose erratum) (electronic).
114. Hliněný, P. and Oum, S. (2007) Finding branch-decompositions and rank-decompositions, submitted. Extended abstract in ESA'07, to appear.
115. Hliněný, P. (2006) Branch-width, parse trees, and monadic second-order logic for matroids. J. Combin. Theory Ser. B, 96 325-351.
116. Hliněný, P. and Seese, D. (2006) Trees, grids, and MSO decidability: From graphs to matroids. Theoret. Comput. Sci., 351, 372-393.
117. Hliněňý, P. (2003) On matroid properties definable in the MSO logic. In Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, Vol. 2747, pp. 470-479. Springer, Berlin.
118. Hliněňý, P. (2007) On some hard problems on matroid spikes. Theory Comput. Syst., to appear.
119. Hliněňý, P. (2006) On matroid representability and minor problems. In Mathematical Foundations of Computer Science, Stará Lesná. Lecture Notes in Computer Science, Vol. 4162, pp. 505-516. Springer, Berlin.
120. Courcelle, B. and Olariu, S. (2000) Upper bounds to the clique width of graphs. Discrete Appl. Math., 101, 77-114.
121. Fellows, M.R., Rosamond, F.A., Rotics, U. and Szeider, S. (2006) Clique-width minimization is NP-hard. Proc. of the 38th Annual ACM Symposium on Theory of Computing, pp. 354-362. ACM Press, New York.
122. Oum, S. and Seymour, P. (2006) Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96, 514-528.
123. Wanke, E. (1994) κ-NLC graphs and polynomial algorithms. Discrete Appl. Math., 54, 251-266.
124. Johansson, Ö. (1998) Clique-decomposition, NLC-decomposition, and modular decomposition - relationships and results for random graphs. Proc. of the 29th Southeastern Int. Conf. Combinatorics, Graph Theory and Computing, Boca Raton, FL, Vol. 132, pp. 39-60.
125. Johansson, Ö. (2001) Graph decomposition using node labels. PhD Thesis, Royal Institute of Technology.
126. Courcelle, B. (2004), Clique-width of countable graphs: A compactness property. Discrete Math. 266, 127-148.
127. Fellows, M.R., Rosamond, F.A., Rotics, U. and Szeider, S. Proving NP-hardness for Clique-width I: Non-approximability of Sequential Clique-width. Report TR05-080, revision 01, Electronic Colloquium on Computational Complexity.
128. Gurski, F. and Wanke, E. (2005) On the relationship between NLC-width and linear NLC-width. Theoret. Comput. Sci., 347, 76-89.
129. Bouchet, A. (1989) Connectivity of isotropic system. In Combinatorial Mathematics: Proc. of the Third International Conference, New York. Annals New York Academic Science, Vol. 555, pp. 81-93. New York Academic Science, New York.
130. van Lint, J.H. and Seidel, J.J. (1966) Equilateral point sets in elliptic geometry. Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math., 28, 335-348.
131. McConnell, R.M. and Spinrad, J. (1999) Modular decomposition and transitive orientation. Discrete Math., 201, 189-241.
132. 5 gem)-free graphs have bounded clique-width. Discrete Appl. Math., 145, 232-241.
133. Cunningham, W.H. (1982) Decomposition of directed graphs. SIAM J. Algebr. Discrete Methods, 3, 214-228.
134. 2) algorithm for undirected split decomposition. J. Algorithms, 16, 145-160.
135. Courcelle, B. (1992) The monadic second-order logic of graphs VII: graphs as relational structures. Theoret. Comput. Sci., 101, 3-33.
136. Courcelle, B. (2006) The monadic second-order logic of graphs XV: On a conjecture by D. Seese. J. Appl. Logic., 4, 79-114.
137. Courcelle, B. and Oum, S. (2007) Vertex-minors, monadic second-order logic, and a conjecture by Seese. J. Combin. Theory Ser. B, 97 91-126.
138. Corneil, D.G. and Rotics, U. (2005) On the relationship between clique-width and treewidth. SIAM J. Comput., 34, 825-847 (electronic).
139. n,n. In Graph-theoretic Concepts in Computer Science Konstanz. Lecture Notes in Computer Science, Vol. 1928, pp. 196 205. Springer, Berlin.
140. Gurski, F. and Wanke, E. (2006) Vertex disjoint paths on clique-width bounded graphs. Theoret. Comput. Sci., 359
141. Gurski, F. and Wanke, E. (2006) Line graphs of bounded clique-width. Discrete Math., doi:10.1016/j.disc.2007.01.020.
142. Corneil, D.G., Lerchs, H. and Burlingham, L.S. (1981) Complement reducible graphs. Discrete Appl. Math., 3, 163-114.
143. Bandelt, H.-J. and Mulder, H.M. (1986) Distance-hereditary graphs. J. Combin. Theory Ser. B, 41, 182-208.
144. Golumbic, M.C. and Rotics, U.(2000) on the clique-width of some perfect graph classes. Int. J. Found. Comput. Sci., 11, 423-443. Selected papers from the Workshop on Theoretical Aspects of Computer Science (WG 99), Part 1 (Ascona).
145. 4's. Int. J. Found. Comput. Sci., 10, 329-348.
146. Boliac, R. and Lozin, V. (2002) On the clique-width of graphs in hereditary classes. Algorithms and Computation: 13th Interntional Symposium, ISAAC 2002, Vancouver, BC, Canada, November, 21-23, Lecture Notes in Computer Science, Vol. 2518, pp. 44-54. Springer.
147. Brandstädt, A., Engelfriet, J., Le, H.-O. and Lozin, V.V. (2006) Clique-width for 4-vertex forbidden subgraphs. Theory Comput. Syst. 39, 561-590.
148. Brandstädt A., Dragan, F.F., Le, H.-O. and Mosca, R. (2005) New graph classes of bounded clique-width. Theory Comput. Syst., 38, 623-645.
149. 4-reducible graphs - a class of uniquely tree-representable graphs. Stud. Appl. Math., 81 79-87.
150. Hoàng, C.T. (1985) Perfect graphs. PhD Thesis, McGill University, Montreal, Canada.
151. 4-tidy graphs. Discrete Math. Theor. Comput. Sci., 1, 17-41 (electronic).
152. 4-partners. Internat. J. Found. Comput. Sci., 10, 103-121.
153. 4's. Graph-theoretic Concepts in Computer Science, Aachen. Lecture Notes in Computer Science, Vol. 1017, pp. 24-36. Springer, Berlin,.
154. Vanherpe, J.-M. (2004) Clique-width of partner-limited graphs. Discrete Math., 276, 63-374; Sixth Int. Conf. Graph Theory
155. Bouchet, A. (1987) Isotropic systems. Eur. J. Combin.. 8, 231-244.
156. Bouchet, A. (1988) Transforming trees by successive local complementations. J. Graph Theory, 12, 195-207.
157. Espelage, W.., Gurski, F. and Wanke, E. (2001) How to solve NP-Hard graph problems on clique-width bounded graphs in polynomial time. Graph-theoretic Concepts in Computer Science Boltenhagen. Lecture Notes in Computer Science, Vol. 2204, pp. 117-128. Springer, Berlin.
158. Kobler, D. and Rotics, U. (2003) Edge dominating set and colorings on graphs with fixed clique-width. Discrete Appl. Math., 126, 197-221.
159. Todinca, I. (2003) Coloring powers of graphs of bounded clique-width. Graph-Theoretic Concepts in Computer Science, October. Lecture Notes in Computer Science, Vol. 2880, pp. 370-382. Springer.
160. Johnson, J.L. (2003) Polynomial time recognition and optimization algorithms on special classes of graphs. PhD Thesis, Vanderbilt University.
161. Espelage, W., Gurski, F. and Wanke, E. (2003) Deciding clique-width for graphs of bounded tree-width. J. Graph Algorithms Appl., 7, 141-180.
162. Dechter, R. and Pearl, J. (1987) Network-based heuristics for constraint-satisfaction problems. Artif. Intell., 34, 1-38.
163. Dechter, R. (1992) Constraint networks. In Encyclopedia of Artificial Intelligence, pp. 276-285. Wiley.
164. Gottlob, G., Leone, N. and Scarcello, F. (2001) The complexity of acyclic conjunctive queries. J. ACM, 48, 431-498.
165. Kolaitis, P.G. and Vardi, M.Y. (2000) Conjunctive-query containment and constraint satisfaction. J. Comput. Syst. Sci., 61, 302-332.
166. Chandra, A.K. and Merlin, P.M. (1977) Optimal implementation of conjunctive queries in relational data bases. Conf. Record of the Ninth Annual ACM Symposium on Theory of Computing, Boulder, CO, pp. 77-90. Assoc. Comput. Mach., New York.
167. Gottlob, G., Leone, N. and Scarcello, F. (2000) A comparison of structural CSP decomposition methods. Artif. Intell., 124, 243-282.
168. Gyssens, M., Jeavons, P.G. and Cohen, D.A. (1994) Decomposing constraint satisfaction problems using database techniques. Artif. Intell., 66, 57-89.
169. Yannakakis, M. (1981) Algorithms for acyclic database schemes. In Very Large Data Bases, Seventh Int. Conf., Cannes, France, September 9-11, pp. 82-94. IEEE Computer Society.
170. Johnson, D.S. (1990) A catalog of complexity classes. Handbook of Theoretical Computer Science, Vol. A, pp. 67-161. Elsevier, Amsterdam.
171. Fagin, R. (1983) Degrees of acyclicity for hypergraphs and relational database schemes. J. Assoc. Comput. Mach., 30, 514-550.
172. Freuder, E.C. (1985) A sufficient condition for backtrack-bounded search. J. Assoc. Comput. Mach., 32, 755-761.
173. Dechter, R. and Pearl, J. (1989) Tree clustering for constraint networks. Artif. Intell., 38, 353-366.
174. Gyssens, M. and Paredaens, J. (1984) A decomposition methodology for cyclic databases. Advances in Database Theory, Vol. 2, pp. 85-122. Plenum Press, New York, NY.
175. Gottlob, G., Leone, N. and Scarcello, F. (2002) Hypertree decompositions and tractable queries. J. Comput. Syst. Sci., 64, 579-627. Special issue on PODS 1999 (Philadelphia, PA).
176. Gottlob, G., Miklós, Z. and Schwentick, T. (2007) Generalized hypertree decompositions: NP-hardness and tractable variants. Proc. of the 26th ACM Symposium on Principles of Database Systems (PODS'07) Beijing, China, June 11-14, pp. 13-22, ACM Press.
177. Chekuri, C. and Rajaraman, A. (2000) Conjunctive query containment revisited. Theoret. Comput. Sci., 239, 211-229.
178. Gottlob, G., Leone, N. and Scarcello, F. (1999) On tractable queries and constraints. DEXA '99: Proc. of the 10th International Conference on Database and Expert Systems Applications, pp. 1-15. Springer, London.
179. Hypertree: Complementary approaches to constraint satisfaction. http://www.dbai.tuwien.ac.at/proj/hypertree/
180. Scarcello, F. (2004) Hypertree decompositions. http://wwwinfo.deis.unical.it/frank/Hypertrees/
181. Bodlaender, H.L. (1996) A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25, 1305-1317.
182. Gottlob, G., Grohe, M., Musliu, N., Samer, M. and Scarcello, F. (2005) Hypertree decompositions: Structure, algorithms, and applications. Kratsch, D. (ed.), Graph-Theoretic Concepts in Computer Science: 31st International Workshop, WG 2005. Lecture Notes in Computer Science, Vol. 3787, pp. 1-15.
183. Gottlob, G. (2005) Computing cores for data exchange: New algorithms and practical solutions. PODS'05: Proc. of the 24th ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, pp. 148-159. ACM Press, New York.
184. Adler, I., Gottlob, G. and Grohe, M. (2007) Hypertree-width and related hypergraph invariants. Eur. J. Combin., doi:10.1016/ j.ejc.2007.04.013.
185. Gottlob, G. and Pichler, R. (2004) Hypergraphs in model checking: acyclicity and hypertree-width versus clique-width. SIAM J. Comput. 33, p351-378 (electronic).
186. Seymour, P.D. and Thomas, R. (1993) Graph searching and a min-max theorem for tree-width. J. Combin. Theory Ser. B, 58, 22-33.
187. Gottlob, G., Leone, N. and Scarcello, F. (2003) Robbers, marshals, and guards: Game theoretic and logical characterizations of hypertree width. J. Comput. Syst. Sci., 66, 775-808. Special issue on PODS 2001 (Santa Barbara, CA).
188. Adler, I. (2004) Marshals, monotone Marshals, and hypertree-width. J. Graph Theory, 47, 275-296.
189. Reed, B.A. (1997) Tree width and tangles: A new connectivity measure and some applications. Surveys in combinatorics, London. London Mathematical Society Lecture Note Series, Vol. 241, pp. 87-162. Cambridge University Press, Cambridge.
190. Hliněný, P. and Whittle. C. (2006) Matroid tree-width. European J. Combin., 27, 1117-1128.
191. Johnson, T., Robertson, N., Seymour, P. and Thomas, R. (2001) Directed tree-width. J. Combin. Theory Ser. B, 82, 138-154.
192. Obdržálek, J. (2006) DAG-with: Connectivity measure for directed graphs. Proc. of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, Miami, FL. pp. 814-821. ACM, New York.
193. Berwanger, D., Dawar, A., Hunter, P. and Kreutzer, S. (2006) DAG-width and parity games. STACS '06, Lecture Notes in Computer Science. Vol. 3884, pp. 524-536. Springer.
194. Hunter, P. and Kreutzer, S. (2007) Digraph measures: Kelly decompositions, games, and orderings. Proc. of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, LA, pp. 637-644. ACM, New York.
195. Hliněný, P. and Obdržálek, J. (2006) Escape-width: Measuring 'width' of digraphs. Presentation at the Sixth Czech-Slovak International Symposium on Combinations, Graph Theory, Algorithms and Applications, Prague, July 10-15.
196. Adler, I. (2007) Directed tree-width examples. J. Comb. Theory Ser. B doi:10.1016/j.jctb.2006.12.006. 97, pp, 718-725.
197. Obdržálek, J. (2006) Algorithmic analysis of parity games. PhD Thesis, University of Edinburgh.
198. Johnson, T., Robertson, N., Seymour, P. and Thomas, R. (2002) Addendum to Directed tree-width. http://www.math.gatech.edu/thomas/PAP/diradd.pdf
199. Makowsky, J.A. and Mariño, J.P. (2003) Tree-width and the monadic quantifier hierarchy. Theoret. Comput. Sci., 303, 157-170. Logic and complexity in computer science (Créteil, 2001).
200. Sankappanavar, H.P. (1978) Decision problems: History and methods. Mathematical Logic, Proc. First Brazillian Conf., State Univ. Campinas Campinas, 1977. Lecture Notes in Pure and Appl. Math. Vol. 39, pp. 241-291. Dekker, New York.
201. Baudisch, A., Seese, D., Tuschik, H.P. and Weese, M. (1980) Decidability and Generalized Quantifiers, Vol. 3, Mathematical Research, Akademie-Verlag, Berlin.
202. Baudisch, A., Seese, D., Tuschik, P. and Weese, M. (1985) Decidability and quantifier-elimination. Model-theoretic Logics, pp. 235-270. Perspect. Math. Logic, Springer, New York.
203. Rabin, M.O. (1969) Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. Soc., 141, 1-35.
204. Shelah, S. (1975) The monadic theory of order. Ann. Math. (2), 102, 379-419.
205. Stupp, J. (1975) The lattice-model is recursive in the original model. Manuscript, The Hebrew University, Jerusalem.
206. Blumensath, A. (2004) Axiomatising tree-interpretable structures. Theory Comput. Syst., 37, 3-27; Symp. Theoretical Aspects of Computer Science (Antibes-Juan les Pins, 2002).
207. Blumensath, A. and Grädel, E. (2000) Automatic structures. In 15th Annual IEEE Symposium on Logic in Computer Science, (Santa Barbara, CA), pp. 51-62. IEEE Comput. Soc. Press, Los Alamitos, CA.
208. Seese, D.G. (1975) Zur Entscheidbarkeit der monadischen Theorie zweiter Stufe haumartiger Graphen. Wiss. Z. Humboldt-Univ. Berlin Math.-Natur. Reihe, 24, 768-772. Theoretischee Kybernetik und mathematische Logik.
209. Seese, D. (1976) Entscheidbarkeits- und Interpretierbarkeitsfragen Monadischer Theorien zweiter Stufe gewisser Klassen von Graphen. PhD Thesis. Humboldt-Universität, Berlin.
210. Seese, D.G. (1979) Some graph-theoretical operations and decidability. Math. Nachr., 87, 15-21.
211. Rautenberg, W. and Hauschild, K. (1971) Interpretierbarkeit und Entscheidbarkeit in der Graphentheorie. I. Z. Math. Logic Grundlagen Math., 17, 47-55.
212. Seese, D. (1986) The structure of the models of decidable monadic theories of graphs. Ann. Pure Appl. Logic, 53, 169-195.
213. Courcelle, B. (2000)The monadic second-order logic of graphs XII: Planar graphs and planar maps. Theoret. Comput. Sci., 237, 1-32.
214. Courcelle, B. (2003) The monadic second-order logic of graphs XIV: uniformly sparse graphs and edge set quantifications. Theoret. Comput. Sci., 299, 1-36.
215. Feferman, S. and Vaught, R.L. (1959) The first order properties of products of algebraic systems. Fund. Math., 47, 57-103.
216. Makowsky, J.A. (2004) Algorithmic uses of the Feferman-Vaught theorem. Ann. Pure Appl. Logic, 126, 159-213.
217. Arnborg, S., Courcelle, B., Proskurowski, A. and Seese, D. (1993) An algebraic theory of graph reduction. J. Assoc. Comput. Mach., 40, 1134-1164.
218. Corneil, D.G., Perl, Y. and Stewart, L.K. (1985) A linear recognition algorithm for cographs. SIAM J. Comput., 14, 926-934.
219. Corneil, D.G., Habib, M., Lanlignel, J.-M., Reed, B. and Rotics, U. (2000) Polynomial time recognition of clique-width ≤ 3 graphs (extended abstract). In Gonnet, Gastón H. (ed.) LATIN 2000: Theoretical Informatics. 4th Latin American Symposium, Punta del Este, Uruguay, April 10-14. Lecture Notes in computer Science, Vol. 1776, 126-134. Springer, Berlin.
220. Hammer, P.L., and Maffray, F. (1990) Completely separable graphs. Discrete Appl. Math., 27, 85-99. Computational algorithms, operations research and computer science (Burnaby, BC, 1987).
221. Damiand, G., Habib, M. and Paul, C. (2001) A simple paradigm for graph recognition: Application to cographs and distance hereditary graphs. Theoret. Comput. Sci., 263, 99-111.
222. Oum, S. (2005) Approximating rank-width and clique-width quickly. In Graph-theoretic concepts in computer science, Metz. Lecture Notes in Computer Science. Vol. 3787, 49-58. Springer.
223. Oum, S. (2005) Graphs of bounded rank-width, PhD Thesis, Princeton University.