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# Planar Graphs and their Properties

A graph 'G' is said to be planar if it can be drawn on a plane or a sphere so that no two edges cross each other at a non-vertex point.

## Example

## Regions

Every planar graph divides the plane into connected areas called regions.

## Example

Degree of a bounded region **r = deg(r)** = Number of edges enclosing the regions **r**.

deg(1) = 3 deg(2) = 4 deg(3) = 4 deg(4) = 3 deg(5) = 8

Degree of an unbounded region **r = deg(r)** = Number of edges enclosing the regions **r**.

deg(R_{1}) = 4 deg(R_{2}) = 6

In planar graphs, the following properties hold good −

**1.**In a planar graph with 'n' vertices, sum of degrees of all the vertices isn ∑ i=1 deg(V_{i}) = 2|E|**2.**According to**Sum of Degrees of Regions**Theorem, in a planar graph with 'n' regions, Sum of degrees of regions is −n ∑ i=1 deg(r_{i}) = 2|E|

Based on the above theorem, you can draw the following conclusions −

In a planar graph,

If degree of each region is K, then the sum of degrees of regions is

K|R| = 2|E|

If the degree of each region is at least K(≥ K), then

K|R| ≤ 2|E|

If the degree of each region is at most K(≤ K), then

K|R| ≥ 2|E|

**Note** − Assume that all the regions have same degree.

**3.** According to **Euler's Formulae** on planar graphs,

If a graph 'G' is a connected planar, then

|V| + |R| = |E| + 2

If a planar graph with 'K' components then

|V| + |R|=|E| + (K+1)

Where, |V| is the number of vertices, |E| is the number of edges, and |R| is the number of regions.

**4. Edge Vertex Inequality**

If 'G' is a connected planar graph with degree of each region at least 'K' then,

|E| ≤ k/k - 2{|v|-2}

You know, |V| + |R| = |E| + 2

K.|R| ≤ 2|E|

K(|E| - |V| + 2) ≤ 2|E|

(K - 2)|E| ≤ K(|V| - 2)

|E| ≤ k/k - 2{|v| - 2}

**5. If 'G' is a simple connected planar graph, then**

|E| ≤ 3|V| − 6 |R| ≤ 2|V| − 4

There exists at least one vertex V ∈ G, such that deg(V) ≤ 5

**6. If 'G' is a simple connected planar graph (with at least 2 edges) and no triangles, then**

|E| ≤ {2|V| – 4}

**7. Kuratowski's Theorem**

A graph 'G' is non-planar if and only if 'G' has a subgraph which is homeomorphic to K_{5} or K_{3,3}.

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