Is the graph isomorphism problem NP-complete?

It is known that the graph isomorphism problem is in the low hierarchy of class NP, which implies that it is not NP-complete unless the polynomial time hierarchy collapses to its second level.

How do you prove that a subgraph isomorphism is NP-complete?

To prove subgraph isomorphism is NP-complete, it must be formulated as a decision problem. The input to the decision problem is a pair of graphs G and H. The answer to the problem is positive if H is isomorphic to a subgraph of G, and negative otherwise.

Is complete graph isomorphic?

Two graphs that are isomorphic must both be connected or both disconnected. Below are two complete graphs, or cliques, as every vertex in each graph is connected to every other vertex in that graph. As a special case of Example 4, Figure 16: Two complete graphs on four vertices; they are isomorphic.

Are two graphs isomorphic NP?

Graph Isomorphism: Two graphs A and B are isomorphic to each other if they have the same number of vertices and edges, and the edge connectivity is retained. There is a bijection between the vertex sets of the graphs A and B.

How do you know if a graph is isomorphic?

You can say given graphs are isomorphic if they have:

  1. Equal number of vertices.
  2. Equal number of edges.
  3. Same degree sequence.
  4. Same number of circuit of particular length.

Why is subgraph isomorphism NP?

First, observe that subgroup isomorphism is in NP, because if we are given a specification of the subgraph of G and the mapping between its vertices and the vertices of H, we can verify in polynomial time that H is indeed isomorphic to the specified subgraph of G.

How do you prove a graph is isomorphic?

What is a complete graph in graph theory?

In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction).

How do you find the isomorphism of two graphs?

If the vertices {V1, V2, .. Vk} form a cycle of length K in G1, then the vertices {f(V1), f(V2),… f(Vk)} should form a cycle of length K in G2. All the above conditions are necessary for the graphs G1 and G2 to be isomorphic, but not sufficient to prove that the graphs are isomorphic.

Is graph coloring NP-complete?

Graph coloring is computationally hard. It is NP-complete to decide if a given graph admits a k-coloring for a given k except for the cases k ∈ {0,1,2} . In particular, it is NP-hard to compute the chromatic number.

How do you solve NP-complete problems?

NP-Completeness

  1. Use a heuristic. If you can’t quickly solve the problem with a good worst case time, maybe you can come up with a method for solving a reasonable fraction of the common cases.
  2. Solve the problem approximately instead of exactly.
  3. Use an exponential time solution anyway.
  4. Choose a better abstraction.