Edge intersection graphs of systems of grid paths with bounded number of bends

We answer some of the questions raised by Golumbic, Lipshteyn and Stern regarding edge intersection graphs of paths on a grid (EPG graphs). We prove that for any d ≥ 4, in order to represent all n vertex graphs with maximum degree d as edge intersection graphs of n paths, a grid of area Θ(n2) is needed. A bend is a turn of a path at a grid point. Let Bk be the class of graphs that have an EPG representation such that each path has at most k bends. We show several results related to the classes Bk; among them we prove that for any odd integer k, Bk $ Bk+1. Lastly, we show that only a very small fraction of all the 2( n 2 ) labeled graphs on n vertices is in Bk.