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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.

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.

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.

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'.

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

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

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}).

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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