Fast fixed-parameter tractable algorithms for nontrivial generalizations of vertex cover

Our goal in this paper is the development of fast algorithms for recognizing general classes of graphs. We seek algorithms whose complexity can be expressed as a linear function of the graph size plus an exponential function of k, a natural parameter describing the class. In particular, we consider the class Wk(G), where for each graph G in Wk(G), the removal of a set of at most k vertices from G results in a graph in the base graph class G. (If G is the class of edgeless graphs, Wk(G) is the class of graphs with bounded vertex cover.)When G is a minor-closed class such that each graph in G has bounded maximum degree, and all obstructions of G (minor-minimal graphs outside G) are connected, we obtain an O((g + k)|V(G)| + (fk)k) recognition algorithm for Wk(G), where g and f are constants (modest and quantified) depending on the class G. If G is the class of graphs with maximum degree bounded by D (not closed under minors), we can still obtain a running time of O(|V(G)|(D + k) + k(D + k)k+3) for recognition of graphs in Wk(G).Our results are obtained by considering bounded-degree minor-closed classes for which all obstructions are connected graphs, and showing that the size of any obstruction for Wk(G) is O(tk7 + t7k2), where t is a bound on the size of obstructions for G. A trivial corollary of this result is an upper bound of (k + 1)(k + 2) on the number of vertices in any obstruction of the class of graphs with vertex cover of size at most k. These results are of independent graph-theoretic interest.