Constant-factor approximation of the domination number in sparse graphs

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. gorithm 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 2 k -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 2 k -independent set A and a k -dominating set D such that | D | = O ( | A | ) , and these sets can be found in linear time. 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.