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&rep=rep1&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!!