FPT algorithms for domination in sparse graphs and beyond

Abstract

We study the parameterized complexity of domination problems on sparse graphs and beyond. The nowhere dense classes of graphs have been proposed as the main model for sparseness that can be utilized algorithmically. The class of d-degenerate graphs is not nowhere dense, yet domination remains fixed-parameter tractable. Both nowhere dense classes of graphs and d-degenerate graph classes are biclique-free classes, meaning there is an integer t such that no graph in the class contains $K_{t,t}$ as a subgraph. In this paper we show that various domination problems are fixed-parameter tractable on biclique-free classes of graphs. Our algorithms are simple and rely on results from extremal graph theory that bound the number of edges in a t-biclique free graph. In particular, the problems k-Dominating Set, Connected k-Dominating Set, Independent k-Dominating Set and Minimum Weight k-Dominating Set are shown to be FPT, when parameterized by $t+k$, on graphs not containing $K_{t,t}$ as a subgraph. With the exception of Connected k-Dominating Set all described algorithms are linear in the size of the input graph.