Edge-intersection graphs of grid paths: The bend-number

Abstract : 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.
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Daniel Heldt, Kolja Knauer, Torsten Ueckerdt. Edge-intersection graphs of grid paths: The bend-number. Discrete Applied Mathematics, Elsevier, 2014, 167, pp.144 - 162. ⟨10.1016/j.dam.2013.10.035⟩. ⟨hal-01457989⟩

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