What is a graph isomorphism?

Question

I am trying to under isomorphism in graphs, and from what I know, if graph A is isomorphic to graph B, then you could basically just rearrange the nodes in A, while keeping the edges connected the same, and get graph B. But I found a question and answer from the internet here:

http://people.math.sfu.ca/~goddyn/Courses/345shutdown/WestSolutions/solutions1.1.pdf

Question 1.1.34

here is a screen shot below, and the three sub graphs shouldn't be isomorphic. For example, the first graph has 2 nodes of degree 3 and the other 2 don't.

Can someone explain this?

Thanks

Answer 1

Each of the three pictures they show are of completely different decompositions. Restrict your attention for the moment to just the far left image.

In this graph, they have colored the edges three ways (in the original picture as thick lines, thin lines, and dashed lines).

If you look at each color class individually, they are all isomorphic to the shape shown below. They are saying that for the peterson graph, you can find a decomposition such that all three subsets of the edges are isomorphic to the same subgraph (which are furthermore symmetric according to rotation), and give three different examples of such decompositions.