In a paper by Zou and Lerman, two examples of graph permutation that correspond to translation and rigid transformation are presented. The first example consists of a periodic lattice graph. It is shown that a given permutation $P$ corresponds to translation in the Euclidean domain.
This example is fairly reasonable and I don't have a problem with it. The authors furthermore give another example where a permutation $P$ would be interpreted as a rigid transformation.
Then the authors state that
The comparison between Figs. a and c makes it clear that the graph and signal permutation corresponds to a Euclidean rigid transformation in the planar representation of the graph.
Unfortunately, this not very clear for me. In fact it is not clear at all. I was wondering if someone can shed some light on it. Thanks!