Independent Set
An Independent Set is a set of vertices (of a graph), such that they're not adjacent to each other in the graph.
A Maximal independent set is the maximum sized independent set.
Here, {1,5} is an independent set.
{1,4,5,6} is Maximal Independent Set.
Clique
Clique is the exact opposite of an Independent set.
Clique is a set of vertices (of a graph), such that they're all adjacent to each other in the graph.
{2,4,5,7} is a clique
Vertex-Cover
Vertex-cover is a set of vertices (of a graph), such that all the edges in the graph are incident on some vertex in it. In other words, the set of vertices that "cover" each edge.
Here {2,3,7} is a vertex-cover.
Source
Now, coming to the question, we can observe some things here:-
- Maximal Independent Set + Minimal Vertex-Cover = All the vertices. (Evident from Image-1 and Image-3)
So, $8+x=20$ $=> x=12$
Option A
More:-
- A set S is independent iff V-S is a vertex-cover.
=> If a set of vertices is vertex-cover (minimal or not) the remaining vertices together form an independent set. And vice versa.
- A set S is independent in G iff S is a clique in G'. Evident from their definitions.