Fixed-parameter algorithms for minor-closed graphs ( of locally bounded treewidth ) ?

Frick and Grohe [7] showed that for each property φ that is definable in firstorder logic, and for each class of minor-closed graphs of locally bounded treewidth, there is an O(n)-time algorithm deciding whether a given graph has property φ. In this paper, we extend this result for fixed-parameter algorithms and show that any minor-closed [contraction-closed] bidimensional parameter which can be computed in polynomial time on graphs of bounded treewidth is also fixed-parameter tractable on general minor-closed graphs [minor-closed class of graphs of locally bounded treewidth]. These parameters include many domination and covering parameters such as vertex cover, feedback vertex set, dominating set, and clique-transversal set. Our algorithm is very simple and its running time is explicit (in contrast to the work of [7]). Along the way, we obtain interesting combinatorial bounds between the aforementioned parameters and the treewidth of the graphs.