Fixed Parameter Algorithms for Minor-Closed Graphs (of Locally Bounded Treewidth)

Abstract

Frick and Grohe showed that for each property phi that is definable in first-order logic, and for each class of minor-closed graphs of locally bounded treewidth, there is an O(n^(1+epsilon))-time algorithm deciding whether a given graph has property phi. 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 Frick and Grohe). Along the way, we obtain interesting combinatorial bounds between the aforementioned parameters and the treewidth of the graphs.