Plane Maps and Planar Graphs: A Mutual Relationship?

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All plane maps can indeed be represented by planar graphs, as each region corresponds to a vertex and each border to an edge. Conversely, every planar graph can be represented as a plane map, adhering to the rules of graph theory. The discussion raises the question of why non-planar graphs cannot be represented as maps, highlighting the limitations of certain graph structures. The relationship between plane maps and planar graphs is foundational in topology and graph theory. Understanding this mutual representation is crucial for further studies in these fields.
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Plane Map <=> Planar Graph ??

Hi,
Wondering if anyone can help me to find some kind of proof that all Plane Maps can be represented by planar graphs and vice versa... for all Planar Graphs exists a Plane Map?

I did a quick search without much luck.

Thanks
 
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Jummeh said:
Hi,
Wondering if anyone can help me to find some kind of proof that all Plane Maps can be represented by planar graphs and vice versa... for all Planar Graphs exists a Plane Map?

I did a quick search without much luck.

Thanks
I can only gave a rough guess, I may not even give an answer relevant to the question, but if I understand you, then each "region" or "country" on the map can be represented by a vertex of the graph, and every "border" between "regions" on the map can be represented by an edge on the graph.
 
yes basically, but why can there not be maps of non-planar graphs?
 
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