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# Independent Vertex Set

Independent sets are represented in sets, in which

there should not be

**any edges adjacent to each other**. There should not be any common vertex between any two edges.there should not be

**any vertices adjacent to each other**. There should not be any common edge between any two vertices.

## Independent Vertex Set

Let 'G' = (V, E) be a graph. A subset of 'V' is called an independent set of 'G' if no two vertices in 'S' are adjacent.

### Example

Consider the following subsets from the above graphs −

S_{1}= {e} S_{2}= {e, f} S_{3}= {a, g, c} S_{4}= {e, d}

Clearly, S_{1} is not an independent vertex set, because for getting an independent vertex set, there should be at least two vertices in the form a graph. But here it is not that case. The subsets S_{2}, S_{3}, and S_{4} are the independent vertex sets because there is no vertex that is adjacent to anyone vertex from the subsets.

## Maximal Independent Vertex Set

Let 'G' be a graph, then an independent vertex set of 'G' is said to be maximal if no other vertex of 'G' can be added to 'S'.

### Example

Consider the following subsets from the above graphs.

S_{1}= {e} S_{2}= {e, f} S_{3}= {a, g, c} S_{4}= {e, d}

S_{2} and S_{3} are maximal independent vertex sets of 'G'. In S_{1} and S_{4}, we can add other vertices; but in S_{2} and S_{3}, we cannot add any other vertex

## Maximum Independent Vertex Set

A maximal independent vertex set of 'G' with a maximum number of vertices is called the maximum independent vertex set.

### Example

Consider the following subsets from the above graph −

S_{1}= {e} S_{2}= {e, f} S_{3}= {a, g, c} S_{4}= {e, d}

Only S_{3} is the maximum independent vertex set, as it covers the highest number of vertices. The number of vertices in a maximum independent vertex set of 'G' is called the independent vertex number of G (β_{2}).

### Example

For the complete graph K_{n}, Vertex covering number = α_{2}= n-1 Vertex independent number = β_{2}= 1 You have α_{2}+ β_{2}= n In a complete graph, each vertex is adjacent to its remaining (n − 1) vertices. Therefore, a maximum independent set of K_{n}contains only one vertex. Therefore, β_{2}=1 and α_{2}=|v| − β_{2}= n-1

**Note** − For any graph 'G' = (V, E)

α

_{2}+ β_{2}= |v|If 'S' is an independent vertex set of 'G', then (V – S) is a vertex cover of G.

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