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Define two operations:

B-operation: When two multi-edges connect a pair of vertices, replace the multi-edges with a single edge connecting the pair of vertices.

C-operation: When one edge connects vertices $u$ and $v$, another edge connects $v$ and $w$ (where $u \ne w$), and there is no other edge incident to $v$, remove the vertex $v$, and replace the two edges with a new edge connecting $u$ and $w$.

How to prove a non-planar graph can't become a graph with only two vertices and a single edge by repeating B and C?

Two examples:

  • $K_3$, let the vertices be called $A_1,A_2,A_3$. Use C on $A_2$ to remove $A_2$, then B on $A_1$ and $A_3$ to remove the redundant edge.

  • $K_4$, since all vertex degrees are 3, no C is possible, and since there are no multi-edges, no B is possible.

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Show that if $G$ reduces to $H$ via one of your operations and $H$ is planar, then $G$ is also planar.

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  • $\begingroup$ So it means B and C operation has closure on planar. And also their reverse action. $\endgroup$ Apr 17, 2022 at 17:09

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