R. Ganian, F. Slivovsky, S. Szeider:

"Meta-kernelization with structural parameters";

Journal of Computer and System Sciences,82(2016), 333 - 346.

Kernelization is a polynomial-time algorithm that reduces an instance of a parameterized problem to a decision-equivalent instance, the kernel, whose size is bounded by a function of the parameter. In this paper we present meta-theorems that provide polynomial kernels for large classes of graph problems parameterized by a structural parameter of the input graph. Let Cbe an arbitrary but fixed class of graphs of bounded rank-width (or, equivalently, of bounded clique-width). We define the C-cover numberof a graph to be the smallest number of modules its vertex set can be partitioned into, such that each module induces a subgraph that belongs to C. We show that each decision problem on graphs which is expressible in Monadic Second Order (MSO) logic has a polynomial kernel with a linear number of vertices when parameterized by the C-cover number. We provide similar results for MSO expressible optimization and modulo-counting problems.

http://dx.doi.org/10.1016/j.jcss.2015.08.003

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