Showing that certain graphs are isomorphic

Question

The above picture shows three graphs. Isn't it clear that all three are isomorphic? I think so, but am concerned that I may be missing something...

Answer 1

Some problems, like this one, are so obviously true that you don't know where to begin proving it. The place to begin is the definitions. What is the definition of "isomorphic"? That there is an isomorphism between them. What is the definition of an isomorphism? It's a function from the vertex set of one graph to the vertex set of another graph, which fulfills some specific properties. Show, possibly by explicitly creating one, that an isomorphism exists, and you're done.

Answer 2

Intuitively: Is there a way of moving around the vertices of the first graph (without adding or eliminating edges) that brings you to the second and third graphs?

Formally: Let $G_1=(V_1,E_1)$ be the first graph, $G_2=(V_2,E_2)$ the second. Is there a bijection $\varphi:V_1\to V_2$ such that $vw$ is an edge of $V_1$ if and only if $\varphi(v)\varphi(w)$ is an edge of $V_2$ (such a function is called an isomorphism)? Is there a similar function with the third graph?

Let $\varphi:V_1\to V_2$ be defined by $\varphi(B)=F, \varphi(C)=G, \varphi(A)=H,\varphi(D)=E$. Is this an isomorphism?