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SIAM Journal on Computing
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ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
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SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
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ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
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Parameterized and Exact Computation
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
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IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
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Proceedings of the forty-second ACM symposium on Theory of computing
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SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Deciding First-Order Properties for Sparse Graphs
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Kernelization hardness of connectivity problems in d-degenerate graphs
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
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Journal of Combinatorial Theory Series B
A simpler algorithm and shorter proof for the graph minor decomposition
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ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
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Parameterized Complexity
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The Multivariate Algorithmic Revolution and Beyond
Kernelization hardness of connectivity problems in d-degenerate graphs
Discrete Applied Mathematics
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We give the first linear kernels for Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a minor. In other words, we give polynomial time algorithms that, for a given H-minor free graph G and positive integer k, output an H-minor free graph G' on O(k) vertices such that G has a (connected) dominating set of size k if and only if G' has. Prior to our work, the only polynomial kernel for Dominating Set on graphs excluding a fixed graph H as a minor was due to Alon and Gutner [ECCC 2008, IWPEC 2009] and to Philip, Raman, and Sikdar [ESA 2009] but the size of their kernel is kc(H), where c(H) is a constant depending on the size of H. Alon and Gutner asked explicitly, whether one can obtain a linear kernel for Dominating Set on H-minor free graphs. We answer this question in affirmative. For Connected Dominating Set no polynomial kernel on H-minor free graphs was known prior to our work. Our results are based on a novel generic reduction rule producing an equivalent instance of the problem with treewidth O(√k). The application of this rule in a divide-and-conquer fashion together with protrusion techniques brings us to linear kernels. As a byproduct of our results we obtain the first subexponential time algorithms for Connected Dominating Set, a deterministic algorithm solving the problem on an n-vertex H-minor free graph in time 2O (√k log k) + nO (1) and a Monte Carlo algorithm of running time 2O (√k) + nO (1). For Dominating Set our results implies a significant simplification and refinement of a 2O (√k)nO (1) algorithm on H minor free graphs due to Demaine et al. [SODA 2003, J. ACM 2005].