Bounding the number of edges between star graphs such that graph is planar

Question

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 ,…,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.

Answer 1

That's the best you can do. Take any planar graph G and construct its face-vertex incidence graph H, whose faces all have 4 edges. Let R be the set of faces of G and construct stars any which way using edges in H. This achieves the bound for bipartite planar graphs.