  In a parameterized problem, every instance I comes with a positive integer k.
The problem is said to admit a polynomial kernel if, in polynomial time, one
can reduce the size of the instance I to a polynomial in k, while preserving
the answer. In this work we give two meta-theorems on kernelzation. The first
theorem says that all problems expressible in Counting Monadic Second Order
Logic and satisfying a coverability property admit a polynomial kernel on
graphs of bounded genus. Our second result is that all problems that have
finite integer index and satisfy a weaker coverability property admit a linear
kernel on graphs of bounded genus. These theorems unify and extend all
previously known kernelization results for planar graph problems.
