number $k\in{\ensuremath{\mathbb{N}}}$. For a subset $V'\subseteq V$ we denote the vertex induced subgraph of $G$ by $G[V']$. Finally, we denote a disjoint union of two sets by ${\mathbin{\dot{\cup}}}$.

Parameterized Complexity
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The concept of parameterized complexity is used to refine the analysis of computational hard problems. It was formally introduced by Downey and Fellows in the 1990s [@DF1992]. Parameterized complexity is motivated by the fact that often a large part of the running time of an algorithm depends not on the instance size but on a problem-specific parameter. A parameterized decision problem is called *fixed-parameter tractable* (FPT) if there exists an algorithm that solves it in time , where  $n$ is the size of the instance and $k$ a problem-specific parameter. Such an algorithm is called an *FPT algorithm*. The function $f$ is typically super-polynomial and it is essential that it depends only on $k$. A standard tool to prove that a parameterized problem is FPT and to design an FPT algorithm is the kernelization technique. A *kernelization* is a polynomial-time algorithm which transforms an instance $(I,k)$ of a parameterized problem into an equivalent instance $(I',k')$ of the same problem such that the size of $I'$ is bounded by $g(k)$ 