Plane Maps and Planar Graphs: A Mutual Relationship?

In summary, a plane map can be represented by a planar graph and vice versa. Each region on the map can be represented by a vertex on the graph, and each border between regions can be represented by an edge on the graph. It is not possible for there to be maps of non-planar graphs.
  • #1
Jummeh
5
0
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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  • #2
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.
 
  • #3
yes basically, but why can there not be maps of non-planar graphs?
 

What is a plane map and a planar graph?

A plane map is a graphical representation of the earth's surface, usually displayed on a flat surface. A planar graph is a mathematical representation of a geometric figure that can be drawn on a plane without any of its edges crossing.

What is the relationship between a plane map and a planar graph?

A plane map can be converted into a planar graph by representing each country, state, or region as a vertex and connecting them with edges to represent the borders between them. Similarly, a planar graph can be converted into a plane map by drawing it on a flat surface.

What are the advantages of using a planar graph to represent a plane map?

Using a planar graph to represent a plane map allows for easier visualization and comparison of data. It also allows for more efficient calculations and analysis of the map's features.

What are some real-world applications of plane maps and planar graphs?

Plane maps and planar graphs are commonly used in geographic information systems (GIS) to analyze and represent data related to land use, transportation networks, and urban planning. They are also used in computer graphics and cartography to create interactive and accurate maps.

What are some limitations of using planar graphs to represent plane maps?

One limitation is that planar graphs can only represent two-dimensional maps, so they cannot accurately represent the curvature of the earth's surface. Additionally, they may not be able to accurately represent complex geographic features and boundaries. Furthermore, converting a plane map into a planar graph may result in some distortion of the data.

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