\[fig:nphardness\], we draw the described configuration. The support line is indicated with dashes. The main line is just below the support line. There are six vertex lines, and six edge lines. That means in the hypergraph $H = (X, F)$, $|X| = 6$ and $|F| = 3$. Since every wireless node on the main line and the vertex lines are distributed uniformly, and the Euclidean distance between parallel lines is always greater than one unit, it is easy to fix the groups of the nodes on the main line and the vertex lines. Moreover, notice the flags on the leftmost and the rightmost vertex lines. These flags help us to fix the flag vertices those contain them.

All fixed nodes are indicated with white circles. The remaining nodes, flag nodes, are indicated with black circles. Now, our task is to find which flag nodes form a collinear group on an edge line.

The edge lines are two symmetrical copies of the edges in $F$ with different colors with respect to the main line. We denote the two copies of an edge $f \in F$ as $b_f$ and $r_f$, for blue and red colors respectively.

If a vertex line that represents 