  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 $k$ such that grid-paths
with at most $k$ 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 $k$ there is a graph with bend-number $k$. 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 $K_{m,n}$ and determine it exactly for some pairs
of $m$ and $n$. Finally, we prove that recognizing single-bend graphs is
NP-complete, providing the first such result in this field.
