Let G be a simple graph with 20 vertices and 100 edges. The size of the minimum vertex cover of G is 8. Then, the size of the maximum indepenĀdent set of G is**(A)** 12**(B)** 8**(C)** Less than 8**(D)** More than 12**Answer:** **(A)****Explanation:** **Background Explanation:****Vertex cover** is a set S of vertices of a graph such that each edge of the graph is incident to at least one vertex of S.**Independent set** of a graph is a set of vertices such that none of the vertices in this set have an edge connecting them i.e. no two are adjacent. A single vertex is an independent set, but we are interested in maximum independent set, that is largest set which is independent set.

**Relation between Independent Set and Vertex Cover :** An interesting fact is, the number of vertices of a graph is equal to its minimum vertex cover number plus the size of a maximum independent set. How? removing all vertices of minimum vertex cover leads to maximum independent set.

So if S is the size of minimum vertex cover of G(V,E) then the size

of maximum independent set of G is |V| – S.

**Solution:**

size of minimum vertex cover = 8

size of maximum independent set = 20 – 8 =12

Therefore, correct answer is (A).

References :

vertex cover

maximum independent set.

This solution is contributed by **Nitika Bansal.**

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