Are there other interpretations of |G| than |V|, that is |V(G)|?
I don't have the book, so I looked through the pages available on Google Scholar. It seems that they use $|G|$ to denote the size of a graph description that is $|V(G)| + |E(G)|$.

For example on page 24 they say that Vertex Cover is solvable in $2^k|G|$ time using bounded search tree method. In fact this method gives $2^kk|V|$ time for graphs with at most $k(|V| - 1)$ edges (it's easy to show that otherwise graph doesn't have Vertex Cover of size $k$). According to this "reverse engineering" $|G|$ corresponds to $k|V|$ that is essentially $|V| + |E|$.