Edge-intersection graphs of grid paths: the bend-number

We investigate edge-intersection graphs of paths in the plane grid regarding a parameter called the bend-number. The bend-number is related to the interval-number and the track-number of a graph. We provide new upper and lower bounds of the bend-number of any given simple graph in terms of the coloring number, edge clique covers and the maximum degree. We show that the bend-number of an outerplanar graph is at most two and that several subclasses of planar graphs have a bend-number of at most three or four. Moreover we determine the bend-numbers of several complete bipartite graphs. Finally, we prove that recognizing single-bend graphs is NP-complete, providing the first such result in this field.