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 \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{O}(n^2 f(k))$\end{document} 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 \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{O}(n^2 f'(k,l))$\end{document} 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).