On <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e694" altimg="si29.svg"><mml:mi>k</mml:mi></mml:math>-bend and monotonic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e699" altimg="si30.svg"><mml:mi>ℓ</mml:mi></mml:math>-bend edge intersection graphs of paths on a grid

If a graph G can be represented by means of paths on grid, such that each vertex corresponds to one path the grid and two vertices are adjacent if only corresponding share edge, then this is called EPG representation representation. A k-bend an in which has at most k bends. The class all graphs have denoted Bk. Bℓm monotonic ℓ-bend representation, i.e. where ascending both columns rows. It trivial Bkm⊆Bk for k. Moreover, it known Bkm⫋Bk, k=1. By investigating Bk-membership Bkm-membership complete bipartite we prove inclusion also proper k∈{2,3,5} k⩾7. In particular, derive necessary conditions membership fulfilled m, n k, m number partition classes graph. We conjecture Bkm⫋Bk holds k∈{4,6}. Furthermore, show Bk⁄⊆B2k−9m k⩾5. This implies restricting shape lead significant increase bends needed So far no bounds amount were known. B1⊆B3m holds, providing first result kind.