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, i.e., every vertex is represented by a grid path and two vertices are adjacent if and only if the two grid paths share at least one grid-edge. The bend-number is the minimum  kk such that grid-paths with at most  kk bends each suffice to represent a given graph. This parameter is related to the interval-number and the track-number of a graph. We show that for every  kk there is a graph with bend-number  kk. Moreover we provide new upper and lower bounds of the bend-number of graphs in terms of degeneracy, treewidth, edge clique covers and the maximum degree. Furthermore we give bounds on the bend-number of  Km,nKm,n and determine it exactly for some pairs of mm and nn. Finally, we prove that recognizing single-bend graphs is NP-complete, providing the first such result in this field.