Abstract
For a given graph property Π (i.e., a collection Π of graphs), the Π-Contraction problem is to determine whether the input graph G can be transformed into a graph satisfying property Π by contracting at most k edges, where k is a parameter. In this paper, we mainly focus on the parameterized complexity of Π-Contraction problems for Π being H-free (i.e., containing no induced subgraph isomorphic to H) for various fixed graphs H.
We show that Clique Contraction (equivalently, P 3 -Free Contraction for connected graphs) is FPT (fixed-parameter tractable) but admits no polynomial kernel unless NP ⊆ coNP/poly, and prove that Chordal Contraction (equivalently, { C l : l ≥ 4 }-Free Contraction) is W[2]-hard. We completely characterize the parameterized complexity of H -Free Contraction for all fixed 3-connected graphs H: FPT but no polynomial kernel unless NP ⊆ coNP/poly if H is a complete graph, and W[2]-hard otherwise. We also show that H -Free Contraction is W[2]-hard whenever H is a fixed cycle C l for some l ≥ 4 or a fixed path P l for some odd l ≥ 5.
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Cai, L., Guo, C. (2013). Contracting Few Edges to Remove Forbidden Induced Subgraphs. 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_10
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DOI: https://doi.org/10.1007/978-3-319-03898-8_10
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