Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond

Abstract

We show that for every fixed j ≥ i ≥ 1, the k -D ominating S et problem restricted to graphs that do not have K ij (the complete bipartite graph on ( i + j ) vertices, where the two parts have i and j vertices, respectively) as a subgraph is fixed parameter tractable (FPT) and has a polynomial kernel. We describe a polynomial-time algorithm that, given a K i,j -free graph G and a nonnegative integer k , constructs a graph H (the “kernel”) and an integer k ' such that (1) G has a dominating set of size at most k if and only if H has a dominating set of size at most k ', (2) H has O (( j + 1) i + 1 k i 2 ) vertices, and (3) k ' = O (( j + 1) i + 1 k i 2 ). Since d -degenerate graphs do not have K d+1,d+1 as a subgraph, this immediately yields a polynomial kernel on O (( d + 2) d +2 k ( d + 1) 2 ) vertices for the k -D ominating S et problem on d -degenerate graphs, solving an open problem posed by Alon and Gutner [Alon and Gutner 2008; Gutner 2009]. The most general class of graphs for which a polynomial kernel was previously known for k -D ominating S et is the class of K h -topological-minor-free graphs [Gutner 2009]. Graphs of bounded degeneracy are the most general class of graphs for which an FPT algorithm was previously known for this problem. K h -topological-minor-free graphs are K i,j -free for suitable values of i,j (but not vice-versa), and so our results show that k -D ominating S et has both FPT algorithms and polynomial kernels in strictly more general classes of graphs. Using the same techniques, we also obtain an O ( jk i ) vertex-kernel for the k -I ndependent D ominating S et problem on K i,j -free graphs.