Graphs with linearly bounded Ramsey numbers
Journal of Combinatorial Theory Series B
Approximation algorithms for NP-complete problems on planar graphs
Journal of the ACM (JACM)
Planar graph coloring with an uncooperative partner
Journal of Graph Theory
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
On forbidden subdivision characterizations of graph classes
European Journal of Combinatorics
Minimum dominating set approximation in graphs of bounded arboricity
DISC'10 Proceedings of the 24th international conference on Distributed computing
Deciding First-Order Properties for Sparse Graphs
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Characterisations and examples of graph classes with bounded expansion
European Journal of Combinatorics
Sparsity: Graphs, Structures, and Algorithms
Sparsity: Graphs, Structures, and Algorithms
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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.