Parameterized complexity of finding small degree-constrained subgraphs

In this article we study the parameterized complexity of problems consisting in finding degree-constrained subgraphs, taking as the parameter the number of vertices of the desired subgraph. Namely, given two positive integers d and k, we study the problem of finding a d-regular (induced or not) subgraph with at most k vertices and the problem of finding a subgraph with at most k vertices and of minimum degree at least d. The latter problem is a natural parameterization of the d-girth of a graph (the minimum order of an induced subgraph of minimum degree at least d). We first show that both problems are fixed-parameter intractable in general graphs. More precisely, we prove that the first problem is W [1]-hard using a reduction from Multi-Color Clique. The hardness of the second problem (for the non-induced case) follows from an easy extension of an already known result. We then provide explicit fixedparameter tractable (FPT) algorithms to solve these problems in graphs with bounded local treewidth and graphs with excluded minors, using a dynamic programming approach. Although these problems can be easily defined in first-order logic, hence by the results of Frick and Grohe [23] are FPT in graphs with bounded local treewidth and graphs with excluded minors, the dependence on k of our algorithms is considerably better than the one following from [23].