On k-bend and monotonic ℓ-bend edge intersection graphs of paths on a grid.

If a graph G can be represented by means of paths on a grid, such that each vertex of G corresponds to one path on the grid and two vertices of G are adjacent if and only if the corresponding paths share a grid edge, then this graph is called EPG and the representation is called EPG representation. A k-bend EPG representation is an EPG representation in which each path has at most k bends. The class of all graphs that have a k-bend EPG representation is denoted by Bk. Bℓm is the class of all graphs that have a monotonic ℓ-bend EPG representation, i.e. an ℓ-bend EPG representation, where each path is ascending in both columns and rows.