Is there a way to convert a nonplanar graph into a planar one, but instead of deleting edges, the only allowed action is to add nodes


1 Answer 1


Logically , that is not Possible.

(Proof 1) Suppose there was a way to add nodes to convert $NP$ a Non-Planar graph into $P$ a Planar graph.
Draw $P$ on a Plane without crossings. Now "delete" the added nodes to get back $NP$ which will still be drawn on a Plane without crossings ! Hence $NP$ is itself a Planar graph !
Contradiction !

(Proof 2) Non-Planar graph has a subgraph which makes it Non-Planar. When we add nodes , that subgraph still Exists (the new graph still has the same subgraph) hence it will still be Non-Planar : We can never get a Planar graph with the addition of nodes.
The subgraphs which may make a given graph Non-Planar :
$K_5$ & $K_{3,3}$


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