# GATE | GATE-CS-2005 | Question 11

• Last Updated : 28 Jun, 2021

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

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