Finding a maximum induced degenerate subgraph faster than 2^n

Abstract

In this paper we study the problem of finding a maximum induced d-degenerate subgraph in a given n-vertex graph from the point of view of exact algorithms. We show that for any fixed d one can find a maximum induced d-degenerate subgraph in randomized (2-eps_d)^n n^O(1) time, for some constant eps_d>0 depending only on d. Moreover, our algorithm can be used to sample inclusion-wise maximal induced d-degenerate subgraphs in such a manner that every such subgraph is output with probability at least (2-eps_d)^-n; hence, we prove that their number is bounded by (2-eps_d)^n.