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Linear time low tree-width partitions and algorithmic consequences

Published: 21 May 2006 Publication History

Abstract

Classes of graphs with bounded expansion have been introduced in [15], [12]. They generalize both proper minor closed classes and classes with bounded degree.For any class with bounded expansion C and any integer p there exists a constant N(C,p) so that the vertex set of any graph G ∈ C may be partitioned into at most N(C,p) parts, any i ≤ p parts of them induce a subgraph of tree-width at most (i-1) [12] (actually, of tree-depth [16] at most i, what is sensibly stronger). Such partitions are central to the resolution of homomorphism problems like restricted homomorphism dualities [14].We give here a simple algorithm to compute such partitions and prove that if we restrict the input graph to some fixed class C with bounded expansion, the running time of the algorithm is bounded by a linear function of the order of the graph (for fixed C and p).This result is applied to get a linear time algorithm for the subgraph isomorphism problem with fixed pattern and input graphs in a fixed class with bounded expansion.More generally, let φ be a first order logic sentence. We prove that any fixed graph property of type "∃X: (|X| ≤ p) ⇿(G[X]=φ)" may be decided in linear time for input graphs in a fixed class with bounded expansion.

References

[1]
N. Alon, R. Yuster, and U. Zwick. Color-coding. J. Assoc. Comput. Mach., 42(4):844--856, 1995.
[2]
D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. J. Symbolic Comput., 9:251--280, 1990.
[3]
B. Courcelle. Graph rewriting: an algebraic and logic approach. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume 2, chapter 5, pages 142--193. Elsevier, Amsterdam, 1990.
[4]
B. Courcelle. The monadic second-order logic of graphs I: recognizable sets of finite graphs. Inform. Comput., 85:12--75, 1990.
[5]
J. Deogun, T. Kloks, D. Kratsch, and H. Muller. On vertex ranking for permutation and other graphs. In Springer, editor, Proceedings if the 11th Annual Symposium on Theoretical Aspects of Computer Science, volume 775 of Lecture Notes in Computer Science, pages 747--758, 1994.
[6]
M. DeVos, G. Ding, B. Oporowski, D. Sanders, B. Reed, P. Seymour, and D. Vertigan. Exluding any graph as a minor allows a low tree-width 2-coloring. Journal of Combinatorial Theory, Series B, 91:25--41, 2004.
[7]
D. Eppstein. Subgraph isomorphism in planar graphs and related problems. In Proc. 6th Symp. Discrete Algorithms, pages 632--640. ACM and SIAM, January 1995.
[8]
D. Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms & Applications, 3(3):1--27, 1999.
[9]
D. Eppstein. Diameter and treewidth in minor-closed graph families. Algorithmica, 27:275--291, 2000. Special issue on treewidth, graph minors, and algorithms.
[10]
R. Halin. S-functions for graphs. J. Geom., 8:171--176, 1976.
[11]
G. Miller, S.-H. Teng, W. Thurston, and S. Vavasis. Geometric separators for finite-element meshes. SIAM J. on Scientific Computing, 19(2):364--386, 1998.
[12]
J. Nešetřil and P. Ossona de Mendez. Grad and classes with bounded expansion I. decompositions. European Journal of Combinatorics, 2005. (submitted).
[13]
J. Nešetřil and P. Ossona de Mendez. Grad and classes with bounded expansion II. algorithmic aspects. European Journal of Combinatorics, 2005. (submitted).
[14]
J. Nešetřil and P. Ossona de Mendez. Grad and classes with bounded expansion III. restricted dualities. European Journal of Combinatorics, 2005. (submitted).
[15]
J. Nešetřil and P. Ossona de Mendez. The grad of a graph and classes with bounded expansion. In A. Raspaud and O. Delmas, editors, 7th International Colloquium on Graph Theory, volume 22 of Electronic Notes in Discrete Mathematics, pages 101--106. Elsevier, 2005.
[16]
J. Nešetřil and P. Ossona de Mendez. Tree depth, subgraph coloring and homomorphism bounds. European Journal of Combinatorics, 2005. (in press).
[17]
J. Nešetřil and S. Poljak. Complexity of the subgraph problem. Comment. Math. Univ. Carol., 26.2:415--420, 1985.
[18]
J. Nešetřil and I. Švejdarová. Diameter of duals are linear. Technical Report 2005-729, KAM-DIMATIA Series, 2005.
[19]
P. Ossona de Mendez. Orientations bipolaires. PhD thesis, Ecole des Hautes Etudes en Sciences Sociales, Paris, 1994.
[20]
J. Plehn and B. Voigt. Finding minimally weighted subgraphs. In Springer-Verlag, editor, Proc. 16th Int. Workshop Graph-Theoretic Concepts in Computer Science, number 484 in Lecture Notes in Computer Science, pages 18--29, 1991.
[21]
N. Robertson and P. Seymour. Graph minors. I. Excluding a forest. J. Combin. Theory Ser. B, 35:39--61, 1983.
[22]
N. Robertson and P. Seymour. Graph minors. XVI. Excluding a non-planar graph. Journal of Combinatorial Theory, Series B, 89(1):43--76, 2003.
[23]
P. Schaffer. Optimal node ranking of trees in linear time. Information Processing Letters, (33):91--96, 1989/90.
[24]
E. Szemerédi. Colloquium at Emory University, Atlanta, GA, April 22. 1994.
[25]
S.-H. Teng. Combinatorial aspects of geometric graphs. Computational Geometry, (9):277--287, 1998.
[26]
R. Thomas. Problem session of the Third Slovene Conference on Graph Theory, Bled, Slovenia. 1995.
[27]
K. Wagner. Über eine Eigenschaft der Ebenen Komplexe. Math. Ann., 114:570--590, 1937.

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    cover image ACM Conferences
    STOC '06: Proceedings of the thirty-eighth annual ACM symposium on Theory of Computing
    May 2006
    786 pages
    ISBN:1595931341
    DOI:10.1145/1132516
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    Published: 21 May 2006

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    Author Tags

    1. bounded expansion
    2. coloration
    3. first order logic
    4. fraternal augmentation
    5. graph minor
    6. subgraph isomorphism
    7. tree-width

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    May 21 - 23, 2006
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    • (2016)Structural sparsityRussian Mathematical Surveys10.1070/RM968871:1(79-107)Online publication date: 20-May-2016
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