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Planar graphs

Maths Applications (Year 12) - Graphs and Networks

Aaron Gibson-John

Planar Graphs

 

A planar graph is a graph that can be drawn on a flat surface (such as a piece of paper) without any of its edges crossing, except at their endpoints. This concept is best demonstrated through a worked example.

 

Example 1: Planar Graph

Consider a simple graph with four vertices (A, B, C, D) and four edges (AB, BC, CD, DA). To determine if this graph is planar, we can try to draw it on a plane without any edge crossings.



Example 2: Non-Planar Graph

Not all graphs are planar. If there are too many edges and too few vertices, then some of the edges will need to intersect. The smallest graph where this happens is K5 – which is a graph with 5 vertices, with one edge between every pair of vertices.


In the next chapter APP3.03.03 we introduce Euler’s formula which can be used to prove whether a graph is planar.

 

Example 3: Demonstrating that a Graph is Planar through Illustration

Below is just one possible solution to showing that the graph on the left is planar.



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