What are Isomorphic Graphs? | Graph Isomorphism, Graph Theory

How do we formally describe two graphs "having the same structure"? The term for this is "isomorphic". Two graphs that have the same structure are called isomorphic, and we'll define exactly what that means with examples in today's video graph theory lesson!

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Two graphs G and H are said to be isomorphic if there exists a bijection, f, from the vertices of G to the vertices of H such that if uv is an edge in G then f(u)f(v) is an edge in H. In other words, the function preserves adjacency and non adjacency. Saying that f is bijective means it is one-to-one (injective) and onto (surjective). Watch the full video for all the details!

SOLUTION TO PRACTICE PROBLEM:

The graphs A and B are isomorphic. Here is an isomorphism, f, between them:

f(v1) = u2
f(v2) = u5
f(v3) = u1
f(v4) = u4
f(v5) = u3

Note that this isomorphism is not unique, so you might have a different one that is also valid. Use the definition of isomorphic graphs to verify your isomorphism.

The graphs C and D are not isomorphic. The graph C has 6 edges and D has only 5. Thus, each of the 6 pairs of adjacent vertices in C could not possibly be matched to 6 pairs of adjacent vertices in D - thus there is no isomorphism.

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I hope you find this video helpful, and be sure to ask any questions down in the comments!

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