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A Linear Kernel for Finding Square Roots of Almost Planar Graphs

Authors Petr A. Golovach, Dieter Kratsch, Daniël Paulusma, Anthony Stewart

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  • planar graphs
  • square roots
  • linear kernel

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Abstract

A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are of distance 2 of each other. The Square Root problem is that of deciding whether a given graph admits a square root. We consider this problem for planar graphs in the context of the "distance from triviality" framework. For an integer k, a planar+kv graph is a graph that can be made planar by the removal of at most k vertices. We prove that the generalization of Square Root, in which we are given two subsets of edges prescribed to be in or out of a square root, respectively, has a kernel of size O(k) for planar+kv graphs, when parameterized by k. Our result is based on a new edge reduction rule which, as we shall also show, has  a wider applicability for the Square Root problem.

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Petr A. Golovach, Dieter Kratsch, Daniël Paulusma, and Anthony Stewart. A Linear Kernel for Finding Square Roots of Almost Planar Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SWAT.2016.4

Author Details

Petr A. Golovach
Dieter Kratsch
Daniël Paulusma
Anthony Stewart

References

  1. Anna Adamaszek and Michal Adamaszek. Uniqueness of graph square roots of girth six. Electronic Journal of Combinatorics, 18, 2011. Google Scholar
  2. Manfred Cochefert, Jean-François Couturier, Petr A. Golovach, Dieter Kratsch, and Daniël Paulusma. Sparse square roots. In Andreas Brandstädt, Klaus Jansen, and Rüdiger Reischuk, editors, Graph-Theoretic Concepts in Computer Science - 39th International Workshop, WG 2013, Lübeck, Germany, June 19-21, 2013, Revised Papers, volume 8165 of Lecture Notes in Computer Science, pages 177-188. Springer, 2013. Google Scholar
  3. Manfred Cochefert, Jean-François Couturier, Petr A. Golovach, Dieter Kratsch, and Daniël Paulusma. Parameterized algorithms for finding square roots. Algorithmica, 74:602-629, 2016. Google Scholar
  4. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  5. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate Texts in Mathematics. Springer, 2012. Google Scholar
  6. Babak Farzad and Majid Karimi. Square-root finding problem in graphs, a complete dichotomy theorem. CoRR, abs/1210.7684, 2012. Google Scholar
  7. Babak Farzad, Lap Chi Lau, Van Bang Le, and Nguyen Ngoc Tuy. Complexity of finding graph roots with girth conditions. Algorithmica, 62:38-53, 2012. Google Scholar
  8. Petr A. Golovach, Dieter Kratsch, Daniël Paulusma, and Anthony Stewart. Squares of low clique number. In 14th Cologne Twente Workshop 2016 (CTW 2016), Gargnano, Italy, June 6-8, 2016, Electronic Notes in Discrete Mathematics, to appear, 2016. Google Scholar
  9. Martin Grötschel, László Lovász, and Alexander Schrijver. Polynomial algorithms for perfect graphs. Annals of Discrete Mathematics, 21:325-356, 1984. Google Scholar
  10. Jiong Guo, Falk Hüffner, and Rolf Niedermeier. A structural view on parameterizing problems: distance from triviality. In Rodney G. Downey, Michael R. Fellows, and Frank K. H. A. Dehne, editors, Parameterized and Exact Computation, First International Workshop, IWPEC 2004, Bergen, Norway, September 14-17, 2004, Proceedings, volume 3162 of Lecture Notes in Computer Science, pages 162-173. Springer, 2004. Google Scholar
  11. Frank Harary, Richard M. Karp, and William T. Tutte. A criterion for planarity of the square of a graph. Journal of Combinatorial Theory, 2:395-405, 1967. Google Scholar
  12. Lap Chi Lau. Bipartite roots of graphs. ACM Transactions on Algorithms, 2:178-208, 2006. Google Scholar
  13. Lap Chi Lau and Derek G. Corneil. Recognizing powers of proper interval, split, and chordal graphs. SIAM Journal on Discrete Mathematics, 18:83-102, 2004. Google Scholar
  14. Van Bang Le, Andrea Oversberg, and Oliver Schaudt. Polynomial time recognition of squares of ptolemaic graphs and 3-sun-free split graphs. Theoretical Computer Science, 602:39-49, 2015. Google Scholar
  15. Van Bang Le, Andrea Oversberg, and Oliver Schaudt. A unified approach for recognizing squares of split graphs. Manuscript, 2015. Google Scholar
  16. Van Bang Le and Nguyen Ngoc Tuy. The square of a block graph. Discrete Mathematics, 310:734-741, 2010. Google Scholar
  17. Van Bang Le and Nguyen Ngoc Tuy. A good characterization of squares of strongly chordal split graphs. Information Processing Letters, 111:120-123, 2011. Google Scholar
  18. Yaw-Ling Lin and Steven Skiena. Algorithms for square roots of graphs. SIAM Journal on Discrete Mathematics, 8:99-118, 1995. Google Scholar
  19. Martin Milanic, Andrea Oversberg, and Oliver Schaudt. A characterization of line graphs that are squares of graphs. Discrete Applied Mathematics, 173:83-91, 2014. Google Scholar
  20. Martin Milanic and Oliver Schaudt. Computing square roots of trivially perfect and threshold graphs. Discrete Applied Mathematics, 161:1538-1545, 2013. Google Scholar
  21. Rajeev Motwani and Madhu Sudan. Computing roots of graphs is hard. Discrete Applied Mathematics, 54:81-88, 1994. Google Scholar
  22. A. Mukhopadhyay. The square root of a graph. Journal of Combinatorial Theory, 2:290-295, 1967. Google Scholar
  23. Nestor V. Nestoridis and Dimitrios M. Thilikos. Square roots of minor closed graph classes. Discrete Applied Mathematics, 168:34-39, 2014. Google Scholar
  24. Ian C. Ross and Frank Harary. The square of a tree. Bell System Technical Journal, 39:641-647, 1960. Google Scholar
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