Removal of subgraphs and perfect matchings in graphs on surfaces

Let (Formula presented.) be the Euler characteristic of a surface (Formula presented.). We characterize the set of graphs (Formula presented.) of order at most 6 which satisfies the following: If (Formula presented.) is a 5-connected graph embedded in a surface (Formula presented.) of face-width at least (Formula presented.) and (Formula presented.) is a subgraph of (Formula presented.) isomorphic to (Formula presented.) such that (Formula presented.) has even order, then (Formula presented.) has a perfect matching. An analogous result on near-perfect matching is given as well. Moreover, we show the following result. Let (Formula presented.) be a 5-connected graph embedded in a surface (Formula presented.) and let (Formula presented.) be connected subgraphs of (Formula presented.) of size at most (Formula presented.) which lie pairwise sufficiently far apart in the face-subdivision of (Formula presented.). We prove that, if the face-width of (Formula presented.) is at least (Formula presented.) and (Formula presented.) satisfies (Formula presented.) for each (Formula presented.) with (Formula presented.), then (Formula presented.) satisfies (Formula presented.) as well, where (Formula presented.) is said to satisfy (Formula presented.) if (Formula presented.) has at most (Formula presented.) components for every (Formula presented.). It is also shown that the distance condition can be considerably weakened when (Formula presented.) triangulates the surface. All of these results generalize previous studies on matching extension in graphs on surfaces.