Matching and l-Subgraph Contractibility to Planar Graphs

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 work with a more general problem called P -RestrictedContract in which S, in addition, is required to satisfy a fixed MSOL formula P (S,G). We give an FPT algorithm in time O(nf(k)) which solves P RestrictedContract, where P (S,G) is (i) inclusion-closed and (ii) inert contraction-closed (where inert edges are the edges non-incident to any inclusion minimal solution S). As a specific example, we can solve the l-subgraph contractibility problem in which the edges of a set S are required to form disjoint connected subgraphs of size at most l. This problem can be solved in time O(nf (k, l)) using the general algorithm. We also show that for l ≥ 2 the problem is NP-complete.