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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. Our classes are of the form Wk(G), graphs that can be formed by augmenting graphs in G by adding at most k vertices (and incident edges). If G is the class of edgeless graphs, Wk(G) is the class of graphs with a vertex cover of size at most k. We describe a recognition algorithm for Wk(G) running in time O((g + k)|V (G)| + (fk)k), where g and f are modest constants depending on the class G, 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. If G is the class of graphs with maximum degree bounded by D (not closed under minors), we can still recognize graphs in Wk(G) in time O(|V (G)|(D + k) + k(D + k)k+3). Our results are obtained by considering minor-closed classes G 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.