We have a set of points $P$ and a set of circles $C$ on a plane. Let $C_i$ be the set of points inside the circle $i\in C$. For no pair of circles $i$ and $j$ we have $C_i = C_j$. We construct a graph $G = (V, E)$ using $P$ and $C$ as following:
Let $V = P$
For each circle $i \in C$ we draw edges between the vertices in $C_i$ as following:
We partition the vertices in $C_i$ into $\lceil \frac{|C_i|}{4} \rceil$ parts with size 4 except for one if the size of $C_i$ is not a multiple of 4. For example, if $C_i = \{a,b,c,d,e,f,g\}$, then we partition $C_i$ as $\{\{a,b,c,d\}, \{e, f, g\}\}$.
Then for each part in this partition, we draw an edge between every vertex. For example, if we partitioned $C_i$ as $\{\{a,b,c,d\}, \{e, f, g\}\}$, then we first draw an edge between every vertex in $\{a,b,c,d\}$, then we draw an edge between every vertex in $\{e, f, g\}$.
I believe that the resulting graph $G$ is always planar, which makes my job much easier for the main problem that I am currently working on. However, I am not sure how I can approach to this problem.
The resulting graph need not be planar.
Let $P=\{a,b,c,d,e,x_1,x_2,x_3\}$, and let $C_i=\{a,b,c,d,e,x_i\}$, $1\le i\le 3$. For $C_1$ we pick the partition $\{\{a,x_1\},\{b,c,d,e\}\}$, for $C_2$ we pick $\{\{b,x_2\},\{a,c,d,e\}\}$, for $C_3$ we pick $\{\{c,d,e,x_3\},\{a,b\}\}$. Then the resulting graph contains $K_5$ and is not planar.