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 B _k. B _ℓ ^m 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. It is trivial that B _k ^m⊆B _k for all k. Moreover, it is known that B _k ^m⫋B _k, for k=1. By investigating the B _k-membership and the B _k ^m-membership of complete bipartite graphs we prove that the inclusion is also proper for k∈{2,3,5} and for k⩾7. In particular, we derive necessary conditions for this membership that have to be fulfilled by m, n and k, where m and n are the number of vertices on the two partition classes of the bipartite graph. We conjecture that B _k ^m⫋B _k holds also for k∈{4,6}. Furthermore, we show that B _k⁄⊆B _2k−9 ^m holds for all k⩾5. This implies that restricting the shape of the paths can lead to a significant increase of the number of bends needed in an EPG representation. So far no bounds on the amount of that increase were known. We prove that B ₁⊆B ₃ ^m holds, providing the first result of this kind.