In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H

${\displaystyle f\colon V(G)\to V(H)}$

such that any two vertices u and v of G are adjacent in G if and only if ${\displaystyle f(u)}$ and ${\displaystyle f(v)}$ are adjacent in H. This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection. If an isomorphism exists between two graphs, then the graphs are called isomorphic and denoted as ${\displaystyle G\simeq H}$. In the case when the bijection is a mapping of a graph onto itself, i.e., when G and H are one and the same graph, the bijection is called an automorphism of G. If a graph is finite, we can prove it to be bijective by showing it is one-one/onto; no need to show both. Graph isomorphism is an equivalence relation on graphs and as such it partitions the class of all graphs into equivalence classes. A set of graphs isomorphic to each other is called an isomorphism class of graphs. The question of whether graph isomorphism can be determined in polynomial time is a major unsolved problem in computer science, known as the Graph Isomorphism problem.

The two graphs shown below are isomorphic, despite their different looking drawings