Table of Contents

## What is a clique independent set?

1 Cliques and Independent Sets. Definition 1. A set of vertices is called independent if no two vertices in the set are adjacent. A set of vertices is called a clique if every two vertices in the set are adjacent.

**Is independent set NP-Complete?**

The independent set decision problem is NP-complete, and hence it is not believed that there is an efficient algorithm for solving it. The maximum independent set problem is NP-hard and it is also hard to approximate.

**Is clique opposite of independent set?**

clique and independent set are not directly complementary graphs of eachother, but the same set of vertices mean two different things based on the complement of the bigger graph..

### What is a clique NP-Complete?

Prerequisite: NP-Completeness. A clique is a subgraph of a graph such that all the vertices in this subgraph are connected with each other that is the subgraph is a complete graph.

**What is clique in algorithm?**

By convention, in algorithm analysis, the number of vertices in the graph is denoted by n and the number of edges is denoted by m. A clique in a graph G is a complete subgraph of G. That is, it is a subset K of the vertices such that every two vertices in K are the two endpoints of an edge in G.

**Is a complete graph a clique?**

A complete graph is often called a clique. The size of the largest clique that can be made up of edges and vertices of G is called the clique number of G.

#### Is clique 3 NP-complete?

a) Prove that CLIQUE-3 is in NP. Verifying each node is in polynomial time since there are a maximum number of edges and nodes in a clique. There also can only be a polynomial number of cliques in a graph due to the limit of 3 edges on each vertex. Therefore, the verifier runs in polynomial time and CLIQUE-3 is in NP.

**Is clique problem NP-complete?**

The clique decision problem is NP-complete (one of Karp’s 21 NP-complete problems). The problem of finding the maximum clique is both fixed-parameter intractable and hard to approximate.

**Is clique decision problem NP-complete?**

The clique decision problem is NP-complete (one of Karp’s 21 NP-complete problems). The problem of finding the maximum clique is both fixed-parameter intractable and hard to approximate. And, listing all maximal cliques may require exponential time as there exist graphs with exponentially many maximal cliques.

## What is clique decision?

In the field of computer science, the clique decision problem is a kind of computation problem for finding the cliques or the subsets of the vertices which when all of them are adjacent to each other are also called complete subgraphs.

**Are the clique and independent set problems NP-complete?**

We know that both the clique and independent set problems are NP-Complete in of themselves. We also know that the verification of this problem, given some “certificate” is in NP.

**Is the independent set NP-complete or NP-hard?**

Therefore, any instance of the independent set problem can be reduced to an instance of the clique problem. Thus, the independent set is NP-Hard. Since the Independent Set problem is both NP and NP-Hard, therefore it is an NP-Complete problem.

### How to prove that the clique problem is NP-hard?

To prove that the clique problem is NP-Hard, we take the help of a problem that is already NP-Hard and show that this problem can be reduced to the Clique problem. For this, we consider the Independent Set problem, which is NP-Complete (and hence NP-Hard ).

**What is an instance of an independent set problem?**

An instance of the problem is an input specified to the problem. An instance of the Independent Set problem is a graph G (V, E) and a positive integer k, and the problem is to check whether an independent set of size k exists in G.