In the following paper by Valiant

http://www.computer.org/csdl/trans/tc/1981/02/06312176.pdf

He shows under theorem 2 (at the bottom of the second page) that any planar graph $G$ of degree 3 or 4 with size $n$ can be embedded in the grid $Gd_{3n,3n}$ (the square grid of size $9n^{2})$. My question is has he made some kind of assumptions on these graphs as just because the number of edges (the size) is $n$ what's to stop us having a loads of isolated vertices which are not able to be embedded? He doesn't seem to mention that the graph is connected or that it contains no isolated vertices.