Vertex Contact Graphs of Paths on a Grid

Abstract

We study Vertex Contact representations of Paths on a Grid (VCPG). In such a representation the vertices of $G$ are represented by a family of interiorly disjoint grid-paths. Adjacencies are represented by contacts between an endpoint of one grid-path and an interior point of another grid-path. Defining $u \rightarrow v$ if the path of $u$ ends on path of $v$ we obtain an orientation on $G$ from a VCPG. To get hand on the bends of the grid path the orientation is not enough. We therefore consider pairs ($\alpha ,\psi$): a 2-orientation $\alpha$ and a flow $\psi$ in the angle graph. The 2-orientation describes the contacts of the ends of a grid-path and the flow describes the behavior of a grid-path between its two ends. We give a necessary and sufficient condition for such a pair $(\alpha ,\psi$) to be realizable as a VCPG.

Using realizable pairs we show that every planar (2, 2)-tight graph admits a VCPG with at most 2 bends per path and that this is tight. Using the same we show that simple planar (2, 1)-sparse graphs have a 4-bend representation and simple planar (2, 0)-sparse graphs have 6-bend representation. We do not believe that the latter two are tight, we conjecture that simple planar (2, 0)-sparse graphs have a 3-bend representation.