R. Belmonte
Parameterized Complexity of Two Edge Contraction Problems with Degree Constraints
Belmonte, R.; Golovach, P.A.; Hof, van 't P.; Paulusma, D.
Authors
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).
Citation
Belmonte, R., Golovach, P., Hof, V. '. P., & Paulusma, D. (2013). Parameterized Complexity of Two Edge Contraction Problems with Degree Constraints. In 8th International Symposium, IPEC 2013, 4-6 September 2013, Sophia Antipolis, France ; revised selected papers (16-27). https://doi.org/10.1007/978-3-319-03898-8_3
Conference Name | 8th International Symposium, IPEC 2013 |
---|---|
Conference Location | Sophia Antipolis, France |
Publication Date | Jan 1, 2013 |
Deposit Date | Dec 20, 2014 |
Publicly Available Date | Jan 15, 2015 |
Pages | 16-27 |
Series Title | Lecture notes in computer science |
Series Number | 8246 |
Series ISSN | 0302-9743,1611-3349 |
Book Title | 8th International Symposium, IPEC 2013, 4-6 September 2013, Sophia Antipolis, France ; revised selected papers. |
DOI | https://doi.org/10.1007/978-3-319-03898-8_3 |
Public URL | https://durham-repository.worktribe.com/output/1153201 |
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Copyright Statement
The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-03898-8_3
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