This seems like it shouldn't be a problem, but here we are.
In 'McKay’s Canonical Graph Labeling Algorithm':
http://www.math.unl.edu/~aradcliffe1/Papers/Canonical.pdf
on page 6, we have figure 1, a simple graph G = (V, E); V = {1,2,3,4,5,6,7,8,9), E = {(1,2), (2,3), (4,5), (5,6), (7,8), (8,9), (1,4), (2,5), (3,6), (4,7), (5,8), (6,9)}. (See link for picture. It's four squares inside a square.)
Later, on page 9, it claims that figure 3 (call it G') is an isomorphism of G. wtf? G'(E) contains (1,8), which isn't in G(E). What have I misunderstood?
Definition of a graph isomorphism:
In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H
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 ƒ(u) and ƒ(v) are adjacent in H.
Both graphs are definitely isomorphic, since they visually have the same shape. More explicitly, one possible edge-preserving bijection $f \colon V(G) \to V(G')$ is given by: \begin{align*} f(1) &= 1 \\ f(2) &= 8 \\ f(3) &= 3 \\ f(4) &= 7 \\ f(5) &= 9 \\ f(6) &= 6 \\ f(7) &= 4 \\ f(8) &= 5 \\ f(9) &= 2 \\ \end{align*} For example, $1$ and $2$ are adjacent in $G$ and $f(1) = 1$ and $f(2) = 8$ are adjacent in $G'$.