Constant-factor approximation of the domination number in sparse graphs

Abstract

The k-domination number of a graph is the minimum size of a set D such that every vertex of G is at distance at most k from D. We give a linear-time constant-factor algorithm for approximation of the k-domination number in classes of graphs with bounded expansion, which include e.g. proper minor-closed graph classes, proper classes closed on topological minors and classes of graphs that can be drawn on a fixed surface with bounded number of crossings on each edge.The algorithm is based on the following approximate min–max characterization. A subset A of vertices of a graph G is d-independent if the distance between each two vertices in A is greater than d. Note that the size of the largest 2k-independent set is a lower bound for the k-domination number. We show that every graph from a fixed class with bounded expansion contains a 2k-independent set A and a k-dominating set D such that |D|=O(|A|), and these sets can be found in linear time.For a fixed value of k, the assumptions on the class can be formulated more precisely in terms of generalized coloring numbers. In particular, for the domination number (k=1), the results hold for all graph classes with arrangeability bounded by a constant.