The extremal function for complete minors
Journal of Combinatorial Theory Series B
Vertex cover: further observations and further improvements
Journal of Algorithms
Infeasibility of instance compression and succinct PCPs for NP
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
On Problems without Polynomial Kernels (Extended Abstract)
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
A quadratic kernel for feedback vertex set
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
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
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Complexity and approximation results for the connected vertex cover problem
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
SODA '10 Proceedings of the twenty-first 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
Sharp tractability borderlines for finding connected motifs in vertex-colored graphs
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Linear time algorithms for finding a dominating set of fixed size in degenerated graphs
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Linear kernels for (connected) dominating set on H-minor-free graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Kernel bounds for path and cycle problems
IPEC'11 Proceedings of the 6th international conference on Parameterized and Exact Computation
Clique cover and graph separation: new incompressibility results
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond
ACM Transactions on Algorithms (TALG)
FPT algorithms for domination in biclique-free graphs
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Finding a maximum induced degenerate subgraph faster than 2n
IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
Parameterized complexity of Min-power multicast problems in wireless ad hoc networks
Theoretical Computer Science
Kernel bounds for path and cycle problems
Theoretical Computer Science
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A graph is d-degenerate if its every subgraph contains a vertex of degree at most d. For instance, planar graphs are 5-degenerate. Inspired by 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 kernelization hardness of problems that include 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 the polynomial hierarchy collapses up to the third level. We prove this using a problem originated from bioinformatics --COLOURFUL GRAPH MOTIF-- analyzed and proved to be NP-hard by Fellows et al. This problem nicely encapsulates the hardness of the connectivity requirement in kernelization. Our technique yields also an alternative proof that, under the same complexity assumption, STEINER TREE does not admit a polynomial kernel. The original proof, via reduction from SET COVER, is due to Dom, Lokshtanov and Saurabh. We extend our analysis by showing that, unless PH = Σp3, 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 Ki,j as a subgraph.