Uniform Kernelization Complexity of Hitting Forbidden Minors

Abstract

The F -M inor -F ree D eletion problem asks, for a fixed set F and an input consisting of a graph G and integer k , whether k vertices can be removed from G such that the resulting graph does not contain any member of F as a minor. At FOCS 2012, Fomin et al. showed that the special case when F contains at least one planar graph has a kernel of size f ( F ) ċ k g ( F ) for some functions f and g . They left open whether this P lanar F -M inor -F ree D eletion problem has kernels whose size is uniformly polynomial, of the form f ( F ) ċ k c for some universal constant c . We prove that some P lanar F -M inor -F ree D eletion problems do not have uniformly polynomial kernels (unless NP ⊆ coNP/poly), not even when parameterized by the vertex cover number. On the positive side, we consider the problem of determining whether k vertices can be removed to obtain a graph of treedepth at most η. We prove that this problem admits uniformly polynomial kernels with O ( k 6 ) vertices for every fixed η.