Graph Theory Questions Medium
The graph isomorphism problem in graph theory is a computational problem that involves determining whether two given graphs are isomorphic or not. In other words, it asks whether two graphs have the same structure, just with different labels assigned to their vertices.
Formally, given two graphs G and H, the graph isomorphism problem seeks to find a bijective mapping (or permutation) between the vertices of G and H, such that for any two vertices u and v in G, there is an edge between them if and only if there is an edge between their corresponding vertices in H.
The problem is to determine whether such a mapping exists or not. If it does, the graphs are considered isomorphic; otherwise, they are non-isomorphic.
The graph isomorphism problem is known to be in the complexity class NP (nondeterministic polynomial time), meaning that a solution can be verified in polynomial time. However, it is still an open question whether the problem is in the class P (polynomial time), which would imply the existence of an efficient algorithm to solve it.
The graph isomorphism problem has important applications in various fields, including computer science, chemistry, and social network analysis. It is also closely related to other graph theoretical problems, such as graph automorphism and graph canonization.