SIAM Journal on Computing
The extremal function for complete minors
Journal of Combinatorial Theory Series B
Vertex cover: further observations and further improvements
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Parameterized Complexity for Domination Problems on Degenerate Graphs
Graph-Theoretic Concepts in Computer Science
Incompressibility through Colors and IDs
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
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Journal of Computer and System Sciences
Journal of Discrete Algorithms
Two Edge Modification Problems without Polynomial Kernels
Parameterized and Exact Computation
A 4k2 kernel for feedback vertex set
ACM Transactions on Algorithms (TALG)
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses
Proceedings of the forty-second ACM symposium on Theory of computing
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Infeasibility of instance compression and succinct PCPs for NP
Journal of Computer and System Sciences
Kernel bounds for disjoint cycles and disjoint paths
Theoretical Computer Science
Preprocessing for treewidth: a combinatorial analysis through kernelization
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Kernelization of packing problems
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Linear kernels for (connected) dominating set on H-minor-free graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Weak compositions and their applications to polynomial lower bounds for kernelization
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Co-nondeterminism in compositions: a kernelization lower bound for a Ramsey-type problem
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
FPT algorithms for connected feedback vertex set
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
Parameterized complexity of eulerian deletion problems
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
Sharp tractability borderlines for finding connected motifs in vertex-colored graphs
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Parameterized Complexity
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A graph is d-degenerate if every subgraph contains a vertex of degree at most d. For instance, planar graphs are 5-degenerate. Inspired by the recent work by Philip, Raman and Sikdar, who have shown the existence of a polynomial kernel for Dominating Set in d-degenerate graphs, we investigate the kernelization complexity of problems that include a connectivity requirement in this class of graphs. Our main contribution is the proof that Connected Dominating Set does not admit a polynomial kernel in d-degenerate graphs for d=2 unless NP@?coNP/poly, which is known to cause a collapse of the polynomial hierarchy to its third level. We prove this using a problem that originates from bioinformatics-Colourful Graph Motif-analysed and proved to be NP-hard by Fellows et al. This problem nicely encapsulates the hardness of the connectivity requirement in kernelization. Our technique also yields an alternative proof that, under the same complexity assumption, Steiner Tree does not admit a polynomial kernel. The original proof, via a reduction from Set Cover, is due to Dom, Lokshtanov and Saurabh. We extend our analysis by showing that, unless NP@?coNP/poly, there do not exist polynomial kernels for Steiner Tree, Connected Feedback Vertex Set and Connected Odd Cycle Transversal in d-degenerate graphs for d=2. On the other hand, we show a polynomial kernel for Connected Vertex Cover in graphs that do not contain the biclique K"i","j as a subgraph. As a d-degenerate graph cannot contain K"d"+"1","d"+"1 as a subgraph, the results holds also for graphs of bounded degeneracy.