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

Abstract

We answer some of the questions raised by Golumbic, Lipshteyn and Stern and obtain some other results about edge intersection graphs of paths on a grid (EPG graphs). We show that for any d ≥ 4 , in order to represent every n vertex graph with maximum degree d as an edge intersection graph of n paths on a grid, a grid of area Θ ( n 2 ) is needed. We also show several results related to the classes B k -EPG, where B k -EPG denotes the class of graphs that have an EPG representation such that each path has at most k bends. In particular, we prove: For a fixed k and a sufficiently large n , the complete bipartite graph K m , n does not belong to B 2 m − 3 -EPG (it is known that this graph belongs to B 2 m − 2 -EPG); for any odd integer k we have B k -EPG ⫋ B k + 1 -EPG; there is no number k such that all graphs belong to B k -EPG; only 2 O ( k n log ( k n ) ) out of all the 2 n 2 labeled graphs with n vertices are in B k -EPG.