Abstract
Given a graph G and an integer k, the objective of the \(\varPi \) -Contraction problem is to check whether there exists at most k edges in G such that contracting them in G results in a graph satisfying the property \(\varPi \). We investigate the problem where \(\varPi \) is ‘H-free’ (without any induced copies of H). It is trivial that \(H\) -free Contraction is polynomial-time solvable if H is a complete graph of at most two vertices. We prove that, in all other cases, the problem is NP-complete. We then investigate the fixed-parameter tractability of these problems. We prove that whenever H is a tree, except for seven trees, \(H\) -free Contraction is W[2]-hard. This result along with the known results leaves behind only three unknown cases among trees.
This work is partly sponsored by SERB (India) grants “Complexity dichotomies for graph modification problems” (SRG/2019/002276), and “Algorithmic study on hereditary graph properties” (MTR/2022/000692), and a public grant overseen by the French National Research Agency as part of the “Investissements d’Avenir” through the IMobS3 Laboratory of Excellence (ANR-10-LABX-0016), and the IDEX-ISITE initiative CAP 20-25 (ANR-16-IDEX-0001). We also acknowledge support of the ANR project GRALMECO (ANR-21-CE48-0004).
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Chakraborty, D., Sandeep, R.B. (2023). Contracting Edges to Destroy a Pattern: A Complexity Study. In: Fernau, H., Jansen, K. (eds) Fundamentals of Computation Theory. FCT 2023. Lecture Notes in Computer Science, vol 14292. Springer, Cham. https://doi.org/10.1007/978-3-031-43587-4_9
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