Document

# A Polynomial Kernel for Line Graph Deletion

## File

LIPIcs.ESA.2020.42.pdf
• Filesize: 0.55 MB
• 15 pages

## Cite As

Eduard Eiben and William Lochet. A Polynomial Kernel for Line Graph Deletion. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 42:1-42:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.42

## Abstract

The line graph of a graph G is the graph L(G) whose vertex set is the edge set of G and there is an edge between e,f ∈ E(G) if e and f share an endpoint in G. A graph is called line graph if it is a line graph of some graph. We study the Line-Graph-Edge Deletion problem, which asks whether we can delete at most k edges from the input graph G such that the resulting graph is a line graph. More precisely, we give a polynomial kernel for Line-Graph-Edge Deletion with O(k⁵) vertices. This answers an open question posed by Falk Hüffner at Workshop on Kernels (WorKer) in 2013.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
• Theory of computation → Graph algorithms analysis
• Theory of computation → Parameterized complexity and exact algorithms
##### Keywords
• Kernelization
• line graphs
• H-free editing
• graph modification problem

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. N. R. Aravind, R. B. Sandeep, and Naveen Sivadasan. Dichotomy results on the hardness of H-free edge modification problems. SIAM J. Discrete Math., 31(1):542-561, 2017. URL: https://doi.org/10.1137/16M1055797.
2. Lowell W. Beineke. Characterizations of derived graphs. Journal of Combinatorial Theory, 9(2):129-135, 1970. URL: https://doi.org/10.1016/S0021-9800(70)80019-9.
3. Hans L Bodlaender, Leizhen Cai, Jianer Chen, Michael R Fellows, Jan Arne Telle, and Dániel Marx. Open problems in parameterized and exact computation-iwpec 2006. UU-CS, 2006, 2006.
4. Leizhen Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett., 58(4):171-176, 1996. URL: https://doi.org/10.1016/0020-0190(96)00050-6.
5. Leizhen Cai and Yufei Cai. Incompressibility of H-free edge modification problems. Algorithmica, 71(3):731-757, 2015. URL: https://doi.org/10.1007/s00453-014-9937-x.
6. Maria Chudnovsky and Paul Seymour. Claw-free graphs. IV. decomposition theorem. Journal of Combinatorial Theory, Series B, 98(5):839-938, 2008. URL: https://doi.org/10.1016/j.jctb.2007.06.007.
7. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
8. Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, Erik Jan van Leeuwen, and Marcin Wrochna. Polynomial kernelization for removing induced claws and diamonds. Theory Comput. Syst., 60(4):615-636, 2017. URL: https://doi.org/10.1007/s00224-016-9689-x.
9. Daniele Giorgio Degiorgi and Klaus Simon. A dynamic algorithm for line graph recognition. In Manfred Nagl, editor, Graph-Theoretic Concepts in Computer Science, 21st International Workshop, WG '95, Aachen, Germany, June 20-22, 1995, Proceedings, volume 1017 of Lecture Notes in Computer Science, pages 37-48. Springer, 1995. URL: https://doi.org/10.1007/3-540-60618-1_64.
10. R. Diestel. Graph Theory, 4th Edition. Springer, 2012.
11. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013.
12. Eduard Eiben, William Lochet, and Saket Saurabh. A polynomial kernel for paw-free editing. CoRR, abs/1911.03683, 2019. URL: http://arxiv.org/abs/1911.03683.
13. Ehab S. El-Mallah and Charles J. Colbourn. The complexity of some edge deletion problems. IEEE Transactions on Circuits and Systems, 35(3):354-362, 1988. URL: https://doi.org/10.1109/31.1748.
14. Paul Erdős and Richard Rado. Intersection theorems for systems of sets. Journal of the London Mathematical Society, 1(1):85-90, 1960.
15. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. URL: https://doi.org/10.1007/3-540-29953-X.
16. Sylvain Guillemot, Frédéric Havet, Christophe Paul, and Anthony Perez. On the (non-)existence of polynomial kernels for P_𝓁-free edge modification problems. Algorithmica, 65(4):900-926, 2013. URL: https://doi.org/10.1007/s00453-012-9619-5.
17. Frank Harary. Graph theory. Addison-Wesley series in mathematics. Addison-Wesley Pub. Co., 1969.
18. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
19. Stefan Kratsch and Magnus Wahlström. Two edge modification problems without polynomial kernels. Discrete Optimization, 10(3):193-199, 2013. URL: https://doi.org/10.1016/j.disopt.2013.02.001.
20. John M. Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci., 20(2):219-230, 1980. URL: https://doi.org/10.1016/0022-0000(80)90060-4.
21. Dániel Marx and R. B. Sandeep. Incompressibility of H-free edge modification problems: Towards a dichotomy. CoRR, abs/2004.11761, 2020. URL: http://arxiv.org/abs/2004.11761.
22. Najiba Sbihi. Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile. Discrete Mathematics, 29(1):53-76, 1980. URL: https://doi.org/10.1016/0012-365X(90)90287-R.
23. Hassler Whitney. Congruent graphs and the connectivity of graphs. American Journal of Mathematics, 54(1):150-168, 1932. URL: http://www.jstor.org/stable/2371086.
24. Mihalis Yannakakis. Edge-deletion problems. SIAM J. Comput., 10(2):297-309, 1981. URL: https://doi.org/10.1137/0210021.