Abstract
A graph is called {claw, diamond}-free if it contains neither a claw (a \(K_{1,3}\)) nor a diamond (a \(K_4\) with an edge removed) as an induced subgraph, or, equivalently, it is a line graph of a triangle-free graph. We consider the parameterized complexity of the {claw, diamond}-free Edge Deletion problem, where given a graph G and a parameter k, the question is whether one can remove at most k edges from G to obtain a {claw, diamond}-free graph. Our main result is that this problem admits a polynomial kernel. We also show that, even on instances with maximum degree 6, the problem is NP-complete and cannot be solved in time \(2^{o(k)}\cdot |V(G)|^{\mathcal {O}(1)}\), assuming the Exponential Time Hypothesis.
The research was supported by Polish National Science Centre grants DEC-2013/11/D/ST6/03073 (Michał Pilipczuk and Marcin Wrochna) and DEC-2012/05/D/ST6/03214 (Marek Cygan and Marcin Pilipczuk). Michał Pilipczuk is currently holding a post-doc position at Warsaw Center of Mathematics and Computer Science. Moreover research of Marcin Pilipczuk was partially supported by the Centre for Discrete Mathematics and its Applications (DIMAP) at the University of Warwick and by Warwick-QMUL Alliance in Advances in Discrete Mathematics and its Applications.
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Notes
- 1.
A more detailed discussion of the relation between these two problems is provided in the conclusions section.
- 2.
The proofs of statements marked with \((\spadesuit )\) are postponed to the full version of the paper [10].
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Cygan, M., Pilipczuk, M., Pilipczuk, M., van Leeuwen, E.J., Wrochna, M. (2016). Polynomial Kernelization for Removing Induced Claws and Diamonds. In: Mayr, E. (eds) Graph-Theoretic Concepts in Computer Science. WG 2015. Lecture Notes in Computer Science(), vol 9224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53174-7_31
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