Site percolation model that cannot be obtained from a bond percolation model

Question

It is easy to obtain a site percolation model from a bond percolation model on a graph $G$ using the covering graph $G_c$ of $G$. I wondered if one can obtain any site percolation model from any site bond and I read in the Geoffrey Grimmett's book that is not true. Nevertheless he does not give any counterexample, and I cannot imagine someone. Can anybody give me a counter example?

Answer 1

It seems as though "covering graph" is just another name for "line graph". As you can find in the "characterization" section of that wiki page, not every graph appears as a line graph of some other graph. A small 5 vertex example is shown there. I am not familiar with percolation, but I think this is basically what you are asking.