Abstract
We study the computational complexity of graph planarization via edge contraction. The problem Contract asks whether there exists a set S of at most k edges that when contracted produces a planar graph. We give an FPT algorithm in time \(\mathcal{O}(n^2 f(k))\) which solves a more general problem P-RestrictedContract in which S has to satisfy in addition a fixed inclusion-closed MSOL formula P.
For different formulas P we get different problems. As a specific example, we study the ℓ-subgraph contractability problem in which edges of a set S are required to form disjoint connected subgraphs of size at most ℓ. This problem can be solved in time \(\mathcal{O}(n^2 f'(k,l))\) using the general algorithm. We also show that for ℓ ≥ 2 the problem is NP-complete. And it remains NP-complete when generalized for a fixed genus (instead of planar graphs).
The first author acknowledges support of Special focus on Algorithmic Foundations of the Internet, NSF grant #CNS-0721113 and mgvis.com http://mgvis.com . The latter three authors acknowledge support of ESF Eurogiga project GraDR as GAČR GIG/11/E023.
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Abello, J., Klavík, P., Kratochvíl, J., Vyskočil, T. (2012). MSOL Restricted Contractibility to Planar Graphs. In: Thilikos, D.M., Woeginger, G.J. (eds) Parameterized and Exact Computation. IPEC 2012. Lecture Notes in Computer Science, vol 7535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33293-7_19
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