for some computable function $g$ and $k'$ is bounded by a function of $k$. The instance $I'$ is said to be a *kernel* of size $g(k)$. Furthermore, a transformation to a different but, loosely speaking, related problem, is called a *generalized kernelization* and a transformed instance a *generalized kernel* (see [@BODLAENDER2009423]). It is widely known that a parameterized problem is fixed-parameter tractable if and only if it admits a (generalized) kernelization. In the remainder of this paper, the kernel size is expressed in terms of the number of vertices. For more background on parameterized complexity the reader is referred to the books of Downey and Fellows [@DF13] and Flum and Grohe [@flumgrohe].

Twin Relation {#subsec:twinrelation}
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Two vertices $v,\, w$ of a graph are called *twins* if $N(v)\setminus\{w\}=N(w)\setminus\{v\}$ holds. Here $v$ and $w$ can be either adjacent (*true twins*) or non-adjacent (*false twins*). It is easy to see that the twin relation (two vertices are related if they are twins) is an equivalence relation. The equivalence classes are called *twin classes* and the partition of the vertices into twin classes is called the *twin partition*.

All vertices of a twin class are either pairwise adjacent or pairwise non-adjacent. Therefore, twin 