1

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 H1, H2,…,Hn 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.

Tomas
  • 57,621
  • 49
  • 238
  • 373
singhsumit
  • 964
  • 1
  • 9
  • 26
  • Nice question, but I think it's belong to http://cstheory.stackexchange.com/ – Saeed Amiri Apr 04 '11 at 09:34
  • it looks good website.. i wasnt aware.. i got this problem as a small problem while m working on some different problem. please provide the exact link if u think it's answered. – singhsumit Apr 04 '11 at 10:23
  • cstheory stack exchange is a site for research level questions in CS, there isn't answer to your specific question in this site, but you can ask your question there to people answer you. – Saeed Amiri Apr 04 '11 at 10:41

1 Answers1

0

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.

qrqwe
  • 299
  • 1
  • 4
  • the arguments are not clear to me. The face-vertex graph H would be bipartite. Then are you forming stars in the bipartite graph or something? Let me restate the problem differently: given a planar bipartite graph you want to decompose vertices of G(which contain star graphs) or want node-disjoint decomposition of G into say 'x' different stars(assuming it exist). so what is the tightest bound on the number of edges in planar bipartite graph(can 'x' play any role in it??). – singhsumit Apr 04 '11 at 13:35
  • @singhsumit I'm working backward by choosing the graph after edges have been added and then picking out the original edges. – qrqwe Apr 04 '11 at 13:41
  • The face-vertex incidence graph is just an easy way to get bipartite planar graphs with the maximum number of edges. – qrqwe Apr 04 '11 at 13:41