# Relationship of 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 (each path is ascending in both columns and rows) ℓ-bend EPG representation. It is known that B_k^m B_k holds for k=1. We prove that B_k^m B_k holds also for k ∈{2, 3, 5} and for k ≥ 7 by investigating the B_k-membership and B_k^m-membership of complete bipartite graphs. 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.

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