Treewidth: Algorithmic Techniques and Results

This paper gives an overview of several results and techniques for graphs algorithms that compute the treewidth of a graph or that solve otherwise intractable problems when restricted graphs with bounded treewidth more efficiently. Also, several results on graph minors are reviewed.

[1]  Torben Hagerup Dynamic Algorithms for Graphs of Bounded Treewidth , 1997, ICALP.

[2]  Bruno Courcelle,et al.  The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues , 1992, RAIRO Theor. Informatics Appl..

[3]  Hans L. Bodlaender,et al.  The hardness of problems on thin colored graphs , 1995 .

[4]  Hans L. Bodlaender,et al.  NC-Algorithms for Graphs with Small Treewidth , 1988, WG.

[5]  Egon Wanke,et al.  Bounded Tree-Width and LOGCFL , 1993, J. Algorithms.

[6]  Robin Thomas,et al.  Call routing and the ratcatcher , 1994, Comb..

[7]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

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

[9]  Paul D. Seymour,et al.  Graph Minors .XIV. Extending an Embedding , 1995, J. Comb. Theory, Ser. B.

[10]  Dimitrios M. Thilikos,et al.  Constructive Linear Time Algorithms for Branchwidth , 1997, ICALP.

[11]  S. Arnborg,et al.  Characterization and recognition of partial 3-trees , 1986 .

[12]  Siddharthan Ramachandramurthi,et al.  The Structure and Number of Obstructions to Treewidth , 1997, SIAM J. Discret. Math..

[13]  Haim Kaplan,et al.  Pathwidth, Bandwidth, and Completion Problems to Proper Interval Graphs with Small Cliques , 1996, SIAM J. Comput..

[14]  Michael R. Fellows,et al.  Obstructions to Within a Few Vertices or Edges of Acyclic , 1995, WADS.

[15]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[16]  Neil Robertson,et al.  Graph Minors .XIII. The Disjoint Paths Problem , 1995, J. Comb. Theory B.

[17]  A. Satyanarayana,et al.  A characterization of partial 3-trees , 1990, Networks.

[18]  Zsolt Tuza,et al.  Narrowness, pathwidth, and their application in natural language processing , 1992, Discret. Appl. Math..

[19]  Paul D. Seymour,et al.  Graph Minors .XII. Distance on a Surface , 1995, J. Comb. Theory, Ser. B.

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

[21]  Bruno Courcelle,et al.  Equivalent definitions of recognizability for sets of graphs of bounded tree-width , 1996, Mathematical Structures in Computer Science.

[22]  Stefan Arnborg,et al.  Linear time algorithms for NP-hard problems restricted to partial k-trees , 1989, Discret. Appl. Math..

[23]  M. Fellows,et al.  Bounded combinatorial width and forbidden substructures , 1995 .

[24]  Craig A. Tovey,et al.  Deterministic Decomposition of Recursive Graph Classes , 1991, SIAM J. Discret. Math..

[25]  Erick Mata-Montero,et al.  Resilience of partial k-tree networks with edge and node failures , 1991, Networks.

[26]  Paul D. Seymour,et al.  Graph minors. VII. Disjoint paths on a surface , 1988, J. Comb. Theory, Ser. B.

[27]  Robin Thomas,et al.  Quickly Excluding a Planar Graph , 1994, J. Comb. Theory, Ser. B.

[28]  Hans L. Bodlaender,et al.  On Linear Time Minor Tests with Depth-First Search , 1993, J. Algorithms.

[29]  A. Asensio Structural and Algorithmic Aspects of Chordal Graph Embeddings , 1996 .

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

[31]  Stefan Arnborg,et al.  Canonical representations of partial 2- and 3-trees , 1992, BIT.

[32]  Jeffrey Scott Vitter,et al.  Dynamic algorithms for optimization problems in bounded tree-width graphs , 1993, IPCO.

[33]  R. Fildes Journal of the Royal Statistical Society (B): Gary K. Grunwald, Adrian E. Raftery and Peter Guttorp, 1993, “Time series of continuous proportions”, 55, 103–116.☆ , 1993 .

[34]  Paul D. Seymour,et al.  Graph Minors: XV. Giant Steps , 1996, J. Comb. Theory, Ser. B.

[35]  Paul D. Seymour,et al.  Graph minors. VIII. A kuratowski theorem for general surfaces , 1990, J. Comb. Theory, Ser. B.

[36]  Stefan Arnborg,et al.  Graph decompositions and tree automata in reasoning with uncertainty , 1993, J. Exp. Theor. Artif. Intell..

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

[38]  Robin Thomas,et al.  Quickly excluding a forest , 1991, J. Comb. Theory, Ser. B.

[39]  Torben Hagerup,et al.  Parallel Algorithms with Optimal Speedup for Bounded Treewidth , 1995, ICALP.

[40]  Eugene L. Lawler,et al.  Linear-Time Computation of Optimal Subgraphs of Decomposable Graphs , 1987, J. Algorithms.

[41]  Eitan M. Gurari,et al.  Improved Dynamic Programming Algorithms for Bandwidth Minimization and the MinCut Linear Arrangement Problem , 1984, J. Algorithms.

[42]  John R. Gilbert,et al.  Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree , 1995, J. Algorithms.

[43]  G. Grätzer,et al.  The free $$ \mathfrak{m} $$ -lattice on the poset H , 1984 .

[44]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs V: On Closing the Gap Between Definability and Recognizability , 1991, Theor. Comput. Sci..

[45]  Haim Kaplan,et al.  Tractability of parameterized completion problems on chordal and interval graphs: minimum fill-in and physical mapping , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[46]  D. Thilikos,et al.  Treewidth and small separators for graphs with small chordality , 1995 .

[47]  Joseph G. Peters,et al.  Regularity and Locality in K-terminal Graphs , 1994, Discret. Appl. Math..

[48]  Konstantin Yu. Gorbunov,et al.  An Estimate of the Tree-Width of a Planar Graph Which Has Not a Given Planar Grid as a Minor , 1998, WG.

[49]  Paul D. Seymour,et al.  Graph Minors: XVII. Taming a Vortex , 1999, J. Comb. Theory, Ser. B.

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

[51]  Bruno Courcelle,et al.  Graph Rewriting: An Algebraic and Logic Approach , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[52]  Hans L. Bodlaender,et al.  Polynomial Algorithms for Graph Isomorphism and Chromatic Index on Partial k-Trees , 1988, J. Algorithms.

[53]  Paul D. Seymour,et al.  Graph minors. I. Excluding a forest , 1983, J. Comb. Theory, Ser. B.

[54]  Hans L. Bodlaender,et al.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC.

[55]  Shin-Ichi Nakano,et al.  Edge-Coloring Partial k-Trees , 1996, J. Algorithms.

[56]  Jan van Leeuwen,et al.  On Interval Routing Schemes and Treewidth , 1995, WG.

[57]  Michael R. Fellows,et al.  Nonconstructive Advances in Polynomial-Time Complexity , 1987, Inf. Process. Lett..

[58]  Hans L. Boblaender Polynomial algorithms for graph isomorphism and chromatic index on partial k -trees , 1990 .

[59]  James B. Saxe,et al.  Dynamic-Programming Algorithms for Recognizing Small-Bandwidth Graphs in Polynomial Time , 1980, SIAM J. Algebraic Discret. Methods.

[60]  Daniel Bienstock,et al.  Graph Searching, Path-Width, Tree-Width and Related Problems (A Survey) , 1989, Reliability Of Computer And Communication Networks.

[61]  Michael R. Fellows,et al.  On Well-Partial-Order Theory and its Application to Combinatorial Problems of VLSI Design , 1989, SIAM J. Discret. Math..

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

[63]  Joost Engelfriet,et al.  Domino Treewidth , 1997, J. Algorithms.

[64]  Hans L. Bodlaender,et al.  On Disjoint Cycles , 1991, Int. J. Found. Comput. Sci..

[65]  Paul D. Seymour,et al.  Graph minors. VI. Disjoint paths across a disc , 1986, J. Comb. Theory, Ser. B.

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

[67]  C. Pandu Rangan,et al.  Treewidth of Circular-Arc Graphs , 1994, SIAM J. Discret. Math..

[68]  David Fernández-Baca,et al.  Parametric Problems on Graphs of Bounded Tree-Width , 1992, J. Algorithms.

[69]  Dieter Kratsch,et al.  Treewidth of Chordal Bipartite Graphs , 1993, J. Algorithms.

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

[71]  Hans L. Bodlaender,et al.  Complexity of Path-Forming Games , 1993, Theor. Comput. Sci..

[72]  John R. Gilbert,et al.  Approximating Treewidth, Pathwidth, and Minimum Elimination Tree Height , 1991, WG.

[73]  Michael R. Fellows,et al.  Fast search algorithms for layout permutation problems , 1991, Integr..

[74]  Stefan Arnborg,et al.  Forbidden minors characterization of partial 3-trees , 1990, Discret. Math..

[75]  Michael R. Fellows,et al.  Fixed-Parameter Tractability and Completeness IV: On Completeness for W[P] and PSPACE Analogues , 1995, Ann. Pure Appl. Log..

[76]  D. Bienstock,et al.  Algorithmic Implications of the Graph Minor Theorem , 1995 .

[77]  David J. Spiegelhalter,et al.  Local computations with probabilities on graphical structures and their application to expert systems , 1990 .

[78]  Paul D. Seymour,et al.  Graph minors. IX. Disjoint crossed paths , 1990, J. Comb. Theory, Ser. B.

[79]  Greg N. Frederickson,et al.  Maintaining Regular Properties Dynamically in k-Terminal Graphs , 1998, Algorithmica.

[80]  Paul D. Seymour,et al.  Graph Minors. XI. Circuits on a Surface , 1994, J. Comb. Theory, Ser. B.

[81]  Detlef Seese,et al.  Monadic Second Order Logic, Tree Automata and Forbidden Minors , 1990, CSL.

[82]  梶谷 洋司,et al.  Minimal Acyclic Forbidden Minors for the Family of Graphs with Bounded Path - Width , 1991 .

[83]  Christos D. Zaroliagis,et al.  Optimal Parallel Shortest Paths in Small Treewidth Digraphs , 1995, ESA.

[84]  Dieter Kratsch,et al.  Treewidth and Pathwidth of Permutation Graphs , 1995, SIAM J. Discret. Math..

[85]  Hans L. Bodlaender,et al.  Reduction Algorithms for Constructing Solutions in Graphs with Small Treewidth , 1996, COCOON.

[86]  Ton Kloks Treewidth of Circle Graphs , 1996, Int. J. Found. Comput. Sci..

[87]  Paul D. Seymour,et al.  Graph minors. III. Planar tree-width , 1984, J. Comb. Theory, Ser. B.

[88]  Jens Lagergren,et al.  Efficient Parallel Algorithms for Graphs of Bounded Tree-Width , 1996, J. Algorithms.

[89]  Michael R. Fellows,et al.  Nonconstructive tools for proving polynomial-time decidability , 1988, JACM.

[90]  Hans L. Bodlaender,et al.  Dynamic Algorithms for Graphs with Treewidth 2 , 1993, WG.

[91]  B. Monien The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete , 1986 .

[92]  Darko Skorin-Kapov,et al.  NC Algorithms for Recognizing Partial 2-Trees and 3-Trees , 1991, SIAM J. Discret. Math..

[93]  Arvind Gupta,et al.  The Complexity of Subgraph Isomorphism Duality Results for Graphs of Bounded Path and Tree Width , 1995 .

[94]  Ivan Hal Sudborough,et al.  The Vertex Separation and Search Number of a Graph , 1994, Inf. Comput..

[95]  Renate Garbe Tree-width and Path-width of Comparability Graphs of interval Orders , 1994, WG.

[96]  Jan Arne Telle,et al.  Efficient Sets in Partial k-Trees , 1993, Discret. Appl. Math..

[97]  David Eppstein,et al.  The Polyhedral Approach to the Maximum Planar Subgraph Problem: New Chances for Related Problems , 1994, GD.

[98]  Naomi Nishimura,et al.  Characterizations of k-terminal flow networks and computing network flows in partial k-trees , 1995, SODA '95.

[99]  Michael R. Fellows,et al.  Fixed-parameter tractability and completeness III: some structural aspects of the W hierarchy , 1993 .

[100]  Robin Thomas,et al.  On the complexity of finding iso- and other morphisms for partial k-trees , 1992, Discret. Math..

[101]  Michael R. Fellows,et al.  On search, decision, and the efficiency of polynomial-time algorithms , 1994, FOCS 1994.

[102]  Stephen T. Hedetniemi,et al.  Linear algorithms on k-terminal graphs , 1987 .

[103]  Jan van Leeuwen,et al.  On Interval Routing Schemes and Treewidth , 1995, Inf. Comput..

[104]  Michael A. Langston,et al.  obstruction Set Isolation for the Gate Matrix Layout Problem , 1994, Discret. Appl. Math..

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

[106]  M. Fellows,et al.  Beyond NP-completeness for problems of bounded width: hardness for the W hierarchy , 1994, Symposium on the Theory of Computing.

[107]  Damon Kaller,et al.  Definability Equals Recognizability of Partial 3-Trees , 1996, WG.

[108]  Paul D. Seymour,et al.  Graph minors. IV. Tree-width and well-quasi-ordering , 1990, J. Comb. Theory, Ser. B.

[109]  Hans L. Bodlaender,et al.  Parallel Algorithms for Treewidth Two , 1997, WG.

[110]  David Fernández-Baca,et al.  Parametric Module Allocation on Partial k-Trees , 1993, IEEE Trans. Computers.

[111]  Paul D. Seymour,et al.  Graph minors. V. Excluding a planar graph , 1986, J. Comb. Theory B.

[112]  Michael R. Fellows,et al.  On search decision and the efficiency of polynomial-time algorithms , 1989, STOC '89.

[113]  Michael R. Fellows,et al.  A Simple Linear-Time Algorithm for Finding Path-Decompositions of Small Width , 1994, Inf. Process. Lett..

[114]  Cyril Gavoille On the Dilation of Interval Routing , 1997, MFCS.

[115]  Klaus Jansen,et al.  Generalized Coloring for Tree-like Graphs , 1992, WG.

[116]  Ton Kloks,et al.  Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs , 1993, J. Algorithms.

[117]  Siddharthan Ramachandramurthi,et al.  Algorithms for VLSI layout based on graph width metrics , 1994 .

[118]  B. D. Fluiter Algorithms for graphs of small treewidth , 1997 .

[119]  Rolf H. Möhring,et al.  The Pathwidth and Treewidth of Cographs , 1993, SIAM J. Discret. Math..

[120]  M. Habib,et al.  Treewidth of cocomparability graphs and a new order-theoretic parameter , 1994 .

[121]  Mikkel Thorup,et al.  All Structured Programs have Small Tree-Width and Good Register Allocation , 1998, Inf. Comput..

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

[123]  Jan Arne Telle,et al.  Practical Algorithms on Partial k-Trees with an Application to Domination-like Problems , 1993, WADS.

[124]  Daniel P. Sanders On Linear Recognition of Tree-Width at Most Four , 1996, SIAM J. Discret. Math..

[125]  Dominique Perrin,et al.  Finite Automata , 1958, Philosophy.

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

[127]  Jens Gustedt,et al.  On the Pathwidth of Chordal Graphs , 1993, Discret. Appl. Math..