I have a graph $G$ which consists only of star graphs. A star graph consists of one central node having edges to every other node in it. Let $H_1, H_2, \ldots, H_n$ be different star graphs of different sizes which are present in $G$. We call the set of all nodes which are centres in any star graph $R$.
Now suppose these star graphs are building edges to other star graphs such that no edge is incident between any nodes in $R$. Then, how many edges exist at maximum between the nodes in $R$ and the nodes which are not in $R$, if the graph should remain planar?
I want the upper bound on the number of such edges. One upper bound that I have in mind is: consider them as bipartite planar graph where $R$ is one set of vertices and rest of the vertices form another set $A$. We are interested in edges between these sets ($R$ and $A$). Since it is planar bipartite, the number of such edges is bounded by twice the number of nodes in $G$.
What I feel is that is there a better bound, maybe twice the nodes in $A$ plus the number of nodes in $R$.
In case you can disprove my intuition, then that would also be good. Hopefully some of you can come up with a good bound along with some relevant arguments.