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On Supergraphs Satisfying CMSO Properties

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Mateus de Oliveira Oliveira. On Supergraphs Satisfying CMSO Properties. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.CSL.2017.33

Abstract

Let CMSO denote the counting monadic second order logic of graphs. We give a constructive proof that for some computable function f, there is an algorithm A that takes as input a CMSO sentence F, a positive integer t, and a connected graph G of maximum degree at most D, and determines, in time f(|F|,t)*2^O(D*t)*|G|^O(t), whether G has a supergraph G' of treewidth at most t such that G' satisfies F. The algorithmic metatheorem described above sheds new light on certain unresolved questions within the framework of graph completion algorithms. In particular, using this metatheorem, we provide an explicit algorithm that determines, in time f(d)*2^O(D*d)*|G|^O(d), whether a connected graph of maximum degree D has a planar supergraph of diameter at most d. Additionally, we show that for each fixed k, the problem of determining whether G has a k-outerplanar supergraph of diameter at most d is strongly uniformly fixed parameter tractable with respect to the parameter d. This result can be generalized in two directions. First, the diameter parameter can be replaced by any contraction-closed effectively CMSO-definable parameter p. Examples of such parameters are vertex-cover number, dominating number, and many other contraction-bidimensional parameters. In the second direction, the planarity requirement can be relaxed to bounded genus, and more generally, to bounded local treewidth.
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• On Supergraphs Satisfying CMSO Properties

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References

1. Karl Abrahamson and Michael Fellows. Finite automata, bounded treewidth, and well-quasiordering. Contemporary Mathematics, 147:539-539, 1993.
2. Isolde Adler, Martin Grohe, and Stephan Kreutzer. Computing excluded minors. In Proc. of SODA 2008, pages 641-650. SIAM, 2008.
3. Mikołaj Bojańczyk and Michal Pilipczuk. Definability equals recognizability for graphs of bounded treewidth. In Proc. of LICS 2016, pages 407-416. ACM, 2016.
4. Mikołaj Bojańczyk and Michal Pilipczuk. Optimizing tree decompositions in MSO. In Proc. of STACS 2017 (To appear), 2017.
5. Nathann Cohen, Daniel Gonçalves, Eunjung Kim, Christophe Paul, Ignasi Sau, Dimitrios M. Thilikos, and Mathias Weller. A polynomial-time algorithm for outerplanar diameter improvement. In Proc. of the 10th International Computer Science Symposium in Russia (CSR 2015), volume 9139 of LNCS, pages 123-142, 2015.
6. H. Comon, M. Dauchet, R. Gilleron, C. Löding, F. Jacquemard, D. Lugiez, S. Tison, and M. Tommasi. Tree automata techniques and applications. Available at http://www.grappa.univ-lille3.fr/tata, 2007. Release October, 12th 2007.
7. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990.
8. Bruno Courcelle and Joost Engelfriet. Graph structure and monadic second-order logic. A language-theoretic approach. HAL, June 14 2012. URL: http://hal.archives-ouvertes.fr/hal-00646514.
9. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
10. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Monographs in Computer Science. Springer, 1999.
11. Michael Elberfeld. Context-free graph properties via definable decompositions. In Proc. of the 25th Conference on Computer Science Logic (CSL 2016), volume 62 of LIPIcs, pages 17:1-17:16, 2016.
12. Michael R. Fellows and Rodney G. Downey. Parameterized computational feasibility. Feasible Mathematics II, 13:219-244, 1995.
13. Michael R. Fellows and Michael A. Langston. On search decision and the efficiency of polynomial-time algorithms. In Proc. of STOC 1989, pages 501-512. ACM, 1989.
14. Jörg Flum, Markus Frick, and Martin Grohe. Query evaluation via tree-decompositions. Journal of the ACM (JACM), 49(6):716-752, 2002.
15. Fedor V. Fomin, Petr Golovach, and Dimitrios M. Thilikos. Contraction obstructions for treewidth. Journal of Combinatorial Theory, Series B, 101(5):302-314, 2011.
16. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Bidimensionality and kernels. In Proc. of SODA 2010, pages 503-510, 2010.
17. Neil Robertson and Paul D. Seymour. Graph minors. XIII. the disjoint paths problem. Journal of combinatorial theory, Series B, 63(1):65-110, 1995.
18. Neil Robertson and Paul D Seymour. Graph minors. XX. Wagner’s conjecture. Journal of Combinatorial Theory, Series B, 92(2):325-357, 2004.
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