Crown Rule Reduction In Parameterized Complexity - Vertex Cover - Notion Question I am reading the paper "Kernelizations for Parameterized Counting Problems", and had a question regarding some of the notation in the paper (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.86.8295&amp;rep=rep1&amp;type=pdf).

In section 4, the author begins by defining the notion of a crown as follows:

Let $\mathcal{G} := (V, E)$ be a graph. A crown in $\mathcal{G}$ is a bipartite subgraph $\mathcal{C} = (I, N(I), F)$ of $\mathcal{G}$ satisfying the following three conditions.


$I$ is an independent set in $\mathcal{G}$ and $N(I)$ is the set of all neighbours of vertices from $I$ in $\mathcal{G}$.
$F$ contains all of the edges of $E$ that connect vertices from $I$ to $N(I)$.
$\mathcal{C}$ has a matching of cardinality $|N(I)|$.


In then goes on to define a few other pieces of notation. Let $\mathcal{G} - \mathcal{C}$ be the graph obtained from $\mathcal{G}$  by deleting all of the vertices in $\mathcal{C}$ and all edges incident to vertices in $\mathcal{C}$. Also, let $S_\mathcal{C}$ be a vertex cover of $\mathcal{C}$ and let $\mathcal{G}' := (V', E')$ be the graph obtained from $\mathcal{G} - \mathcal{C}$ by deleting all edges covered by vertices in $S_\mathcal{C}$.

What confuses me here is the difference between $\mathcal{G} - \mathcal{C}$ and $\mathcal{G}'$. Since the vertex cover contains only vertices in $I$ or $N(I)$, when removing $\mathcal{C}$ from $\mathcal{G}$, haven't we already removed all of the edges that can be covered by vertices in $S_\mathcal{C}$? It would seem to me that the edges in $\mathcal{G} - \mathcal{C}$ do not have any vertices that are incident, making it identical to $\mathcal{G}'$. Where is the error in my reasoning/understanding?

Thanks!!
I agree with the OP: $\mathcal{G} - \mathcal{C} = \mathcal{G}'$.

I assume the author of the paper wanted to define $\mathcal{G}'$ as $\mathcal{G} - S_{\mathcal{C}}$. With this modification, the next sentence in the paper makes sense:


  One can easily see that $\#vc(\mathcal{G}',k-|S_{\mathcal{C}}|)$ equals the number of size $k$ vertex covers $S$ in $\mathcal{G}$ with $S \supseteq S_{\mathcal{C}}$.


Here, $\#vc(\mathcal{G},k)$ denotes the number of size $k$ vertex covers in $\mathcal{G}$.