Parameterized complexity of finding regular induced subgraphs

The r-Regular Induced Subgraph problem asks, given a graph G and a non-negative integer k, whether G contains an r-regular induced subgraph of size at least k, that is, an induced subgraph in which every vertex has degree exactly r. In this paper we examine its parameterization k-Sizer-Regular Induced Subgraph with k   as parameter and prove that it is W[1]W[1]-hard. We also examine the parameterized complexity of the dual parameterized problem, namely, the k-Almostr-Regular Graph problem, which asks for a given graph G and a non-negative integer k whether G can be made r-regular by deleting at most k   vertices. For this problem, we prove the existence of a problem kernel of size O(kr(r+k)2)O(kr(r+k)2).