Plane Maps and Planar Graphs: A Mutual Relationship?

  • Context: Graduate 
  • Thread starter Thread starter Jummeh
  • Start date Start date
  • Tags Tags
    Graph Map Plane
Click For Summary
SUMMARY

All Plane Maps can be represented by planar graphs, and conversely, every planar graph corresponds to a plane map. This relationship is established through the representation of regions as vertices and borders as edges. The discussion highlights the fundamental nature of planar graphs in depicting geographical maps, emphasizing that non-planar graphs cannot be accurately represented as plane maps due to their inherent complexity.

PREREQUISITES
  • Understanding of planar graphs and their properties
  • Familiarity with graph theory terminology
  • Knowledge of vertex and edge representation in graphs
  • Basic concepts of geographical mapping
NEXT STEPS
  • Research the properties of planar graphs in graph theory
  • Study the Euler's formula for planar graphs
  • Explore the concept of graph embeddings and their applications
  • Learn about non-planar graphs and their characteristics
USEFUL FOR

Mathematicians, computer scientists, and anyone interested in graph theory and its applications in geographical mapping and spatial analysis.

Jummeh
Messages
5
Reaction score
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
 
Mathematics news on Phys.org
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?
 

Similar threads

High School 4-colours theorem can be proved visually
  • · Replies 21 ·
Replies
21
Views
2K
Undergrad Conventionality when Graphing Transformations.
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Graph Theory: (proof)conditions for Euler Circuit in Digraph
  • · Replies 1 ·
Replies
1
Views
4K
Graduate How Does U(1) Double-Cover SO(2) for a Specified Angle?
  • · Replies 2 ·
Replies
2
Views
3K
MHB Understanding Symmetry Maps and Rotations in the Plane
  • · Replies 17 ·
Replies
17
Views
3K
Graduate Physics and Integer Computation with Eisenstein Integers
  • · Replies 1 ·
Replies
1
Views
2K
Show plane curve can be described with graph @ tangent point
  • · Replies 6 ·
Replies
6
Views
2K
High School Mapping plane/set into/onto itself (What exactly does this mean?)
  • · Replies 2 ·
Replies
2
Views
16K
Almost All Graphs are Non-Planar
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Projection of 3D Plane from a certain perspective on the 2D Plane
  • · Replies 2 ·
Replies
2
Views
4K