Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
List edge and list total colourings of multigraphs
Journal of Combinatorial Theory Series B
Polynomial-time data reduction for dominating set
Journal of the ACM (JACM)
Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size
SIAM Journal on Computing
Planar Graphs of Odd-Girth at Least $9$ are Homomorphic to the Petersen Graph
SIAM Journal on Discrete Mathematics
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Linear problem kernels for planar graph problems with small distance property
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
Nonblocker: parameterized algorithmics for minimum dominating set
SOFSEM'06 Proceedings of the 32nd conference on Current Trends in Theory and Practice of Computer Science
On the independence number of graphs with maximum degree 3
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
Linear problem kernels for NP-hard problems on planar graphs
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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Perhaps the best known kernelization result is the kernel of size 335k for the Planar Dominating Set problem by Alber et al. [1], later improved to 67k by Chen et al. [5]. This result means roughly, that the problem of finding the smallest dominating set in a planar graph is easy when the optimal solution is small. On the other hand, it is known that Planar Dominating Set parameterized by k′=|V|−k (also known as Planar Nonblocker) has a kernel of size 2k′. This means that Planar Dominating Set is easy when the optimal solution is very large. We improve the kernel for Planar Nonblocker to $\frac{7}{4}k'$. This also implies that Planar Dominating Set has no kernel of size at most $(\frac{7}{3}-\epsilon)k$, for any ε0, unless P=NP. This improves the previous lower bound of (2−ε)k of [5]. Both of these results immediately generalize to H-minor free graphs (without changing the constants). In our proof of the bound on the kernel size we use a variant of the discharging method (used e.g. in the proof of the four color theorem). We give some arguments that this method is natural in the context of kernelization and we hope it will be applied to get improved kernel size bounds for other problems as well. As a by-product we show a result which might be of independent interest: every n-vertex graph with no isolated vertices and such that every pair of degree 1 vertices is at distance at least 5 and every pair of degree 2 vertices is at distance at least 2 has a dominating set of size at most $\frac{3}7n$.