Abstract
Motivated by recent results of Mathieson and Szeider (J. Comput. Syst. Sci. 78(1): 179–191, 2012), we study two graph modification problems where the goal is to obtain a graph whose vertices satisfy certain degree constraints. The Regular Contraction problem takes as input a graph G and two integers d and k, and the task is to decide whether G can be modified into a d-regular graph using at most k edge contractions. The Bounded Degree Contraction problem is defined similarly, but here the objective is to modify G into a graph with maximum degree at most d. We observe that both problems are fixed-parameter tractable when parameterized jointly by k and d. We show that when only k is chosen as the parameter, Regular Contraction becomes W[1]-hard, while Bounded Degree Contraction becomes W[2]-hard even when restricted to split graphs. We also prove both problems to be NP-complete for any fixed d ≥ 2. On the positive side, we show that the problem of deciding whether a graph can be modified into a cycle using at most k edge contractions, which is equivalent to Regular Contraction when d = 2, admits an O(k) vertex kernel. This complements recent results stating that the same holds when the target is a path, but that the problem admits no polynomial kernel when the target is a tree, unless NP ⊆ coNP/poly (Heggernes et al., IPEC 2011).
This work has been supported by the Research Council of Norway (197548/F20), the European Research Council (267959), EPSRC (EP/G043434/1) and the Royal Society (JP100692).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Asano, T., Hirata, T.: Edge-contraction problems. J. Comput. Syst. Sci. 26(2), 197–208 (1983)
Brouwer, A.E., Veldman, H.J.: Contractibility and NP-completeness. J. Graph Theory 11, 71–79 (1987)
Cai, L.: Parameterized complexity of cardinality constrained optimization problems. The Computer Journal 51(1), 102–121 (2008)
Diestel, R.: Graph Theory, Electronic Edition. Springer-Verlag (2005)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)
Fellows, M.R., Hermelin, D., Rosamond, F., Vialette, S.: On the parameterized complexity of multiple-interval problems. Theor. Comp. Sci. 410, 53–61 (2009)
Golovach, P.A., van ’t Hof, P., Paulusma, D.: Obtaining planarity by contracting few edges. Theor. Comp. Sci. 476, 38–46 (2013)
Golovach, P.A., Kamiński, M., Paulusma, D., Thilikos, D.M.: Increasing the minimum degree of a graph by contractions. Theor. Comp. Sci. 481, 74–84 (2013)
Heggernes, P., van ’t Hof, P., Lévêque, B., Lokshtanov, D., Paul, C.: Contracting graphs to paths and trees. Algorithmica (to appear) doi:10.1007/s00453-012-9670-2
Heggernes, P., van ’t Hof, P., Lokshtanov, D., Paul, C.: Obtaining a bipartite graph by contracting few edges. In: FSTTCS 2011, LIPIcs, vol. 13, pp. 217–228 (2011)
Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comp. System Sci. 20, 219–230 (1980)
Marx, D., O’Sullivan, B., Razgon, I.: Finding small separators in linear time via treewidth reduction. In: ACM Trans. Algorithms (to appear), Manuscript available at http://www.cs.bme.hu/~dmarx/papers/marx-tw-reduction-talg.pdf
Mathieson, L., Szeider, S.: Editing graphs to satisfy degree constraints: A parameterized approach. J. Comput. Syst. Sci. 78, 179–191 (2012)
Moser, H., Thilikos, D.M.: Parameterized complexity of finding regular induced subgraphs. J. Discr. Algorithms 7, 181–190 (2009)
Yannakakis, M.: Edge-deletion problems. SIAM J. Comput. 10(2), 297–309 (1981)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this paper
Cite this paper
Belmonte, R., Golovach, P.A., van ’t Hof, P., Paulusma, D. (2013). Parameterized Complexity of Two Edge Contraction Problems with Degree Constraints. In: Gutin, G., Szeider, S. (eds) Parameterized and Exact Computation. IPEC 2013. Lecture Notes in Computer Science, vol 8246. Springer, Cham. https://doi.org/10.1007/978-3-319-03898-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-03898-8_3
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-03897-1
Online ISBN: 978-3-319-03898-8
eBook Packages: Computer ScienceComputer Science (R0)