"**No. of edges is 100**" (This information is not needed for solving this question).

28 votes

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 independent set of $G$ is:

- $12$
- $8$
- less than $8$
- more than $12$

55 votes

Best answer

Vertex cover: A set of vertices such that each edge of the graph is incident to at least one vertex of the set.

Therefore, removing all the vertices of the vertex cover from the graph results in an isolated graph and the same set of nodes would be the independent set in the original graph.

Size of minimum vertex cover $= 8$

Size of maximum independent set $= 20 - 8 =12$

Therefore, correct answer would be (**A**).

Reference :- http://mathworld.wolfram.com/MaximumIndependentVertexSet.html

45

a set of vertices $X \subseteq V(G)$ of a graph $G$ is independent if and only if its complement $V(G) \setminus X$ is a vertex cover. ( $V(G) \setminus X$ is complement of $X$ )

(Because if $X$ is VC, then for every edge atleast one endpoint will be in $X$. Removing $X$ from $V$ we get all nonadjacent vertices.)

Complement of vertex cover is Independent set and vice versa.

If that vertex cover is minimum, then complement is Maximum Independent set.

(Because if $X$ is VC, then for every edge atleast one endpoint will be in $X$. Removing $X$ from $V$ we get all nonadjacent vertices.)

Complement of vertex cover is Independent set and vice versa.

If that vertex cover is minimum, then complement is Maximum Independent set.

3 votes

$minimum\ vertex\ cover(\beta)+maximum\ independent\ set(\alpha)=n(no.\ of\ vertices)$

$8+x=20 \implies x=20-8 = 12$

$answer\ is\ A)12$

$8+x=20 \implies x=20-8 = 12$

$answer\ is\ A)12$

0 votes

**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).

Reference : https://www.geeksforgeeks.org/gate-gate-cs-2005-question-11/