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Grad and classes with bounded expansion. II: Algorithmic aspects. (English) Zbl 1185.05131
Summary: Classes of graphs with bounded expansion have been introduced in [(1) J. Nešetřil and P. Ossona de Mendez, The grad of a graph and classes with bounded expansion, in: A. Raspaud, The grad of a graph and classes with bounded expansion. A. Raspaud (ed.) et al., 7th international colloquium on graph theory, Hyeres, France, September 12–16, 2005. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 22, 101–106 (2005; Zbl 1182.05102), (2) Grad and classes with bounded expansion I. Decompositions, European Journal of Combinatorics (2005) (submitted for publication)]. They generalize classes with forbidden topological minors (i.e. classes of graphs having no subgraph isomorphic to the subdivision of some graph in a forbidden family), and hence both proper minor closed classes and classes with bounded degree. For any class with bounded expansion $$\mathcal C$$ and any integer $$p$$ there exists a constant $$N({\mathcal C},p)$$ so that the vertex set of any graph $$G\in{\mathcal C}$$ may be partitioned into at most $$N({\mathcal C},p)$$ parts, any $$i\leq p$$ parts of them induce a subgraph of tree-width at most $$(i-1)$$ [(2)] (actually, of tree-depth [J. Nešetřil and P. Ossona de Mendez, Eur. J. Comb. 27, No. 6, 1022–1041 (2006; Zbl 1089.05025)] at most $$i$$, which is sensibly stronger). Such partitions are central to the resolution of homomorphism problems like restricted homomorphism dualities [J. Nešetřil and P. Ossona de Mendez, Grad and classes with bounded expansion III. Restricted dualities, European Journal of Combinatorics (2005) (submitted for publication)]. We give here a simple algorithm for computing such partitions and prove that if we restrict the input graph to some fixed class $$\mathcal C$$ with bounded expansion, the running time of the algorithm is bounded by a linear function of the order of the graph (for fixed $$\mathcal C$$ 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 $$\phi$$ be a first-order logic sentence. We prove that any fixed graph property of type
$\text{}\exists X:(|X|\leq p)\wedge (G[X]\vDash\phi)\text{''}$
may be decided in linear time for input graphs in a fixed class with bounded expansion. We also show that for fixed $$p$$, computing the distances between two vertices up to distance $$p$$ may be performed in constant time per query after a linear time preprocessing.
Also, extending several earlier results, we show that a class of graphs has sublinear separators if it has sub-exponential expansion. This result is best possible in general.

##### MSC:
 05C83 Graph minors 05C75 Structural characterization of families of graphs 05C85 Graph algorithms (graph-theoretic aspects)
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##### References:
 [1] N. Alon, P.D. Seymour, R. Thomas, A separator theorem for graphs with excluded minor and its applications, in: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, 1990, pp. 293-299 [2] Alon, N.; Yuster, R.; Zwick, U., Color-coding, Journal of the association for computing machinery, 42, 4, 844-856, (1995) · Zbl 0885.68116 [3] Chrobak, M.; Eppstein, D., Planar orientations with low out-degree and compaction of adjacency matrices, Theoretical computer science, 86, 243-266, (1991) · Zbl 0735.68015 [4] Coppersmith, D.; Winograd, S., Matrix multiplication via arithmetic progressions, Journal of symbolic computation, 9, 251-280, (1990) · Zbl 0702.65046 [5] Courcelle, B., Graph rewriting: an algebraic and logic approach, (), 142-193, (Chapter 5) · Zbl 0900.68282 [6] Courcelle, B., The monadic second-order logic of graphs I: recognizable sets of finite graphs, Information computation, 85, 12-75, (1990) · Zbl 0722.03008 [7] Deogun, J.S.; Kloks, T.; Kratsch, D.; Muller, H., On vertex ranking for permutation and other graphs, (), 747-758 · Zbl 0941.05516 [8] DeVos, M.; Ding, G.; Oporowski, B.; Sanders, D.P.; Reed, B.; Seymour, P.D.; Vertigan, D., Excluding any graph as a minor allows a low tree-width 2-coloring, Journal of combinatorial theory. series B, 91, 25-41, (2004) · Zbl 1042.05036 [9] Eppstein, D., Subgraph isomorphism in planar graphs and related problems, (), 632-640 · Zbl 0858.05075 [10] Eppstein, D., Subgraph isomorphism in planar graphs and related problems, Journal of graph algorithms & applications, 3, 3, 1-27, (1999) · Zbl 0949.05055 [11] Eppstein, D., Diameter and treewidth in minor-closed graph families (treewidth, graph minors, and algorithms), Algorithmica, 27, 275-291, (2000), (special issue) · Zbl 0963.05128 [12] Gilbert, J.R.; Hutchinson, J.P.; Tarjan, R.E., A separator theorem for graphs of bounded genus, Journal of algorithms, 5, 375-390, (1984) [13] Halin, R., $$S$$-functions for graphs, Journal of geometry, 8, 171-176, (1976) · Zbl 0339.05108 [14] Kostochka, A.V.; Melnikov, L.S., On bounds of the bisection width of cubic graphs, (), 151-154 · Zbl 0773.05069 [15] Lipton, R.; Tarjan, R.E., A separator theorem for planar graphs, SIAM journal on applied mathematics, 36, 2, 177-189, (1979) · Zbl 0432.05022 [16] Miller, G.L.; Teng, S.-H.; Thurston, W.; Vavasis, S.A., Geometric separators for finite-element meshes, SIAM journal on scientific computing, 19, 2, 364-386, (1998) · Zbl 0914.65123 [17] Nešetřil, J.; Ossona de Mendez, P., Grad and classes with bounded expansion I. decompositions, European journal of combinatorics, 29, 3, 760-776, (2008) · Zbl 1156.05056 [18] J. Nešetřil, P. Ossona de Mendez, Grad and classes with bounded expansion III. Restricted dualities, European Journal of Combinatorics (2005) (submitted for publication) [19] Nešetřil, J.; Ossona de Mendez, P., The Grad of a graph and classes with bounded expansion, (), 101-106 · Zbl 1182.05102 [20] Nešetřil, J.; Ossona de Mendez, P., Linear time low tree-width partitions and algorithmic consequences, (), 391-400 · Zbl 1301.05268 [21] Nešetřil, J.; Ossona de Mendez, P., Tree depth, subgraph coloring and homomorphism bounds, European journal of combinatorics, 27, 6, 1022-1041, (2006) · Zbl 1089.05025 [22] Nešetřil, J.; Poljak, S., Complexity of the subgraph problem, Commentationes mathematicae universitatis carolinae, 26, 2, 415-420, (1985) · Zbl 0571.05050 [23] J. Nešetřil, I. Švejdarová, Diameter of duals are linear, Technical Report 2005-729, KAM-DIMATIA Series, Journal of Graph Theory (2005) · Zbl 1139.05318 [24] P. Ossona de Mendez, Orientations bipolaires, Ph.D. Thesis, Ecole des Hautes Etudes en Sciences Sociales, Paris, 1994 [25] Plehn, J.; Voigt, B., Finding minimally weighted subgraphs, (), 18-29 · Zbl 0768.68170 [26] Plotkin, S.; Rao, S.; Smith, W.D., Shallow excluded minors and improved graph decomposition, () · Zbl 0867.05069 [27] Robertson, N.; Seymour, P.D., Graph minors. I. excluding a forest, Journal of combinatorial theory. series B, 35, 39-61, (1983) · Zbl 0521.05062 [28] Robertson, N.; Seymour, P.D., Graph minors. XVI. excluding a non-planar graph, Journal of combinatorial theory. series B, 89, 1, 43-76, (2003) · Zbl 1023.05040 [29] Schaffer, P., Optimal node ranking of trees in linear time, Information processing letters, 33, 91-96, (1989-90) · Zbl 0683.68038 [30] Teng, S.-H., Combinatorial aspects of geometric graphs, Computational geometry, 9, 277-287, (1998) · Zbl 0894.68157 [31] R. Thomas, Problem Session of the Third Slovene Conference on Graph Theory, Bled, Slovenia, 1995 [32] Wagner, K., Über eine eigenschaft der ebenen komplexe, Mathematische annalen, 114, 570-590, (1937) · JFM 63.0550.01
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