Abstract
Tree-depth is an important graph parameter which arose in the study of sparse graph classes. We present a dynamic data structure for representing a graph G with tree-depth at most D. The structure allows addition and removal of edges and vertices under assumption that the resulting graph still has tree-depth at most D, in time bounds depending only on D. A tree-depth decomposition of the graph is maintained explicitly.
This makes the data structure useful for dynamization of static algorithms for graphs with bounded tree-depth. As an example application, we give a dynamic data structure for MSO property testing.
Supported by KONTAKT II LH12095 and SVV 267313.
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References
Kapron, B.M., King, V., Mountjoy, B.: Dynamic graph connectivity in polylogarithmic worst case time. In: Khanna, S. (ed.) SODA, pp. 1131–1142. SIAM (2013)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to algorithms. The MIT Press (2009)
Sleator, D.D., Endre Tarjan, R.: A data structure for dynamic trees. Journal of Computer and System Sciences 26, 362–391 (1983)
Dvořák, Z., Tůma, V.: A dynamic data structure for counting subgraphs in sparse graphs. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 304–315. Springer, Heidelberg (2013)
Courcelle, B.: The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)
Kreutzer, S., Tazari, S.: On brambles, grid-like minors, and parameterized intractability of monadic second-order logic. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 354–364. Society for Industrial and Applied Mathematics, Philadelphia (2010)
Kreutzer, S., Tazari, S.: Lower bounds for the complexity of monadic second-order logic. In: 2010 25th Annual IEEE Symposium on Logic in Computer Science (LICS), pp. 189–198. IEEE (2010)
Dvořák, Z., Král’, D., Thomas, R.: Deciding first-order properties for sparse graphs. In: FOCS, pp. 133–142. IEEE Computer Society (2010)
Grohe, M., Kreutzer, S., Siebertz, S.: Deciding first-order properties of nowhere dense graphs. CoRR abs/1311.3899 (2013)
Nešetřil, J., Ossona de Mendez, P.: Sparsity: Graphs, Structures, and Algorithms, vol. 28. Springer (2012)
Nešetřil, J., Ossona de Mendez, P.: Tree-depth, subgraph coloring and homomorphism bounds. European Journal of Combinatorics 27, 1022–1041 (2006)
Lampis, M.: Model checking lower bounds for simple graphs. CoRR abs/1302.4266 (2013)
Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. In: Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science, pp. 215–224. IEEE (2002)
Lampis, M.: Algorithmic meta-theorems for restrictions of treewidth. Algorithmica 64, 19–37 (2012)
Gajarsky, J., Hlineny, P.: Faster Deciding MSO Properties of Trees of Fixed Height, and Some Consequences. In: D’Souza, D., Kavitha, T., Radhakrishnan, J. (eds.) IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012), Germany. Leibniz International Proceedings in Informatics (LIPIcs), vol. 18, pp. 112–123. Dagstuhl, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2012)
Nešetřil, J., Ossona de Mendez, P.: Grad and classes with bounded expansion I. decompositions. European Journal of Combinatorics 29, 760–776 (2008)
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Dvořák, Z., Kupec, M., Tůma, V. (2014). A Dynamic Data Structure for MSO Properties in Graphs with Bounded Tree-Depth. In: Schulz, A.S., Wagner, D. (eds) Algorithms - ESA 2014. ESA 2014. Lecture Notes in Computer Science, vol 8737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_28
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DOI: https://doi.org/10.1007/978-3-662-44777-2_28
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