Planar Graph in Graph Theory | Planar Graph Example

Types of Graphs-

 

Before you go through this article, make sure that you have gone through the previous article on various Types of Graphs in Graph Theory.

 

We have discussed-

  • A graph is a collection of vertices connected to each other through a set of edges.
  • The study of graphs is known as Graph Theory.

 

 

In this article, we will discuss about Planar Graphs.

 

Planar Graph-

 

A planar graph may be defined as-

 

In graph theory,

Planar graph is a graph that can be drawn in a plane such that none of its edges cross each other.

 

Planar Graph Example-

 

The following graph is an example of a planar graph-

 

 

Here,

  • In this graph, no two edges cross each other.
  • Therefore, it is a planar graph.

 

Regions of Plane-

 

The planar representation of the graph splits the plane into connected areas called as Regions of the plane.

 

Each region has some degree associated with it given as-

  • Degree of Interior region = Number of edges enclosing that region
  • Degree of Exterior region = Number of edges exposed to that region

 

Example-

 

Consider the following planar graph-

 

 

Here, this planar graph splits the plane into 4 regions- R1, R2, R3 and R4 where-

  • Degree (R1) = 3
  • Degree (R2) = 3
  • Degree (R3) = 3
  • Degree (R4) = 5

 

Planar Graph Chromatic Number-

 

  • Chromatic Number of any planar graph is always less than or equal to 4.
  • Thus, any planar graph always requires maximum 4 colors for coloring its vertices.

 

Planar Graph Properties-

 

Property-01:

 

In any planar graph, Sum of degrees of all the vertices = 2 x Total number of edges in the graph

 

 

Property-02:

 

In any planar graph, Sum of degrees of all the regions = 2 x Total number of edges in the graph

 

 

Special Cases

 

Case-01:

 

In any planar graph, if degree of each region is K, then-

 

K x |R| = 2 x |E|

 

Case-02:

 

In any planar graph, if degree of each region is at least K (>=K), then-

 

K x |R| <= 2 x |E|

 

Case-03:

 

In any planar graph, if degree of each region is at most K (<=K), then-

 

K x |R| >= 2 x |E|

 

 

Property-03:

 

If G is a connected planar simple graph with ‘e’ edges, ‘v’ vertices and ‘r’ number of regions in the planar representation of G, then-

 

r = e – v + 2

 

This is known as Euler’s Formula.

It remains same in all the planar representations of the graph.

 

Property-04:

 

If G is a planar graph with k components, then-

 

r = e – v + (k + 1)

 

Also Read- Bipartite Graph

 

PRACTICE PROBLEMS BASED ON PLANAR GRAPH IN GRAPH THEORY-

 

Problem-01:

 

Let G be a connected planar simple graph with 25 vertices and 60 edges. Find the number of regions in G.

 

Solution-

 

Given-

  • Number of vertices (v) = 25
  • Number of edges (e) = 60

 

By Euler’s formula, we know r = e – v + 2.

 

Substituting the values, we get-

Number of regions (r)

= 60 – 25 + 2

= 37

 

Thus, Total number of regions in G = 37.

 

Problem-02:

 

Let G be a planar graph with 10 vertices, 3 components and 9 edges. Find the number of regions in G.

 

Solution-

 

Given-

  • Number of vertices (v) = 10
  • Number of edges (e) = 9
  • Number of components (k) = 3

 

By Euler’s formula, we know r = e – v + (k+1).

 

Substituting the values, we get-

Number of regions (r)

= 9 – 10 + (3+1)

= -1 + 4

= 3

 

Thus, Total number of regions in G = 3.

 

Problem-03:

 

Let G be a connected planar simple graph with 20 vertices and degree of each vertex is 3. Find the number of regions in G.

 

Solution-

 

Given-

  • Number of vertices (v) = 20
  • Degree of each vertex (d) = 3

 

Calculating Total Number Of Edges (e)-

 

By sum of degrees of vertices theorem, we have-

 

Sum of degrees of all the vertices = 2 x Total number of edges

Number of vertices x Degree of each vertex = 2 x Total number of edges

20 x 3 = 2 x e

∴ e = 30

 

Thus, Total number of edges in G = 30.

 

Calculating Total Number Of Regions (r)-

 

By Euler’s formula, we know r = e – v + 2.

 

Substituting the values, we get-

Number of regions (r)

= 30 – 20 + 2

= 12

 

Thus, Total number of regions in G = 12.

 

Problem-04:

 

Let G be a connected planar simple graph with 35 regions, degree of each region is 6. Find the number of vertices in G.

 

Solution-

 

Given-

  • Number of regions (n) = 35
  • Degree of each region (d) = 6

 

Calculating Total Number Of Edges (e)-

 

By sum of degrees of regions theorem, we have-

 

Sum of degrees of all the regions = 2 x Total number of edges

Number of regions x Degree of each region = 2 x Total number of edges

35 x 6 = 2 x e

∴ e = 105

 

Thus, Total number of edges in G = 105.

 

Calculating Total Number Of Vertices (v)-

 

By Euler’s formula, we know r = e – v + 2.

 

Substituting the values, we get-

35 = 105 – v + 2

∴ v = 72

 

Thus, Total number of vertices in G = 72.

 

Problem-05:

 

Let G be a connected planar graph with 12 vertices, 30 edges and degree of each region is k. Find the value of k.

 

Solution-

 

Given-

  • Number of vertices (v) = 12
  • Number of edges (e) = 30
  • Degree of each region (d) = k

 

Calculating Total Number Of Regions (r)-

 

By Euler’s formula, we know r = e – v + 2.

 

Substituting the values, we get-

Number of regions (r)

= 30 – 12 + 2

= 20

 

Thus, Total number of regions in G = 20.

 

Calculating Value Of k-

 

By sum of degrees of regions theorem, we have-

 

Sum of degrees of all the regions = 2 x Total number of edges

Number of regions x Degree of each region = 2 x Total number of edges

20 x k = 2 x 30

∴ k = 3

 

Thus, Degree of each region in G = 3.

 

Problem-06:

 

What is the maximum number of regions possible in a simple planar graph with 10 edges?

 

Solution-

 

In a simple planar graph, degree of each region is >= 3.

So, we have 3 x |R| <= 2 x |E|.

 

Substituting the value |E| = 10, we get-

3 x |R| <= 2 x 10

|R| <= 6.67

|R| <= 6

 

Thus, Maximum number of regions in G = 6.

 

Problem-07:

 

What is the minimum number of edges necessary in a simple planar graph with 15 regions?

 

Solution-

 

In a simple planar graph, degree of each region is >= 3.

So, we have 3 x |R| <= 2 x |E|.

 

Substituting the value |R| = 15, we get-

3 x 15 <= 2 x |E|

|E| >= 22.5

|E| >= 23

 

Thus, Minimum number of edges required in G = 23.

 

To gain better understanding about Planar Graphs in Graph Theory,

Watch this Video Lecture

 

Next Article- Euler Graph

 

Get more notes and other study material of Graph Theory.

Watch video lectures by visiting our YouTube channel LearnVidFun.

Summary
Planar Graph in Graph Theory | Planar Graph Example
Article Name
Planar Graph in Graph Theory | Planar Graph Example
Description
Planar Graph in Graph Theory- A planar graph is a graph that can be drawn in a plane such that none of its edges cross each other. Planar Graph Example, Properties & Practice Problems are discussed.
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Gate Vidyalay
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