FPT Algorithms for Plane Completion Problems

The Plane Subgraph (resp. Topological Minor) Completion problem asks, given a (possibly disconnected) plane (multi)graph Γ and a connected plane (multi)graph ∆, whether it is possible to add edges in Γ without violating the planarity of its embedding so that it contains some subgraph (resp. topological minor) that is topologically isomorphic to ∆. We give FPT algorithms that solve both problems in f(|E(∆)|) · |E(Γ)|2 steps. Moreover, for the Plane Subgraph Completion problem we show that f(k) = 2O(k log k). 1998 ACM Subject Classification G.2.2 Graph Algorithms