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Journal of Graph Algorithms and Applications
Article . 2015 . Peer-reviewed
License: CC BY
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Article . 2015
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Article . 2021
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Vertex Contact Representations of Paths on a Grid

Vertex contact representations of paths on a grid
Authors: Nieke Aerts; Stefan Felsner;

Vertex Contact Representations 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 on a square grid. Adjacencies are represented by contacts between an endpoint of one grid-path and an interior point of another grid-path. Defining u → v if the path of u ends on the path of v, we obtain an orientation on G from a VCPG. To control the bends of the grid paths the orientation is not enough. We therefore consider pairs (α,ψ): a 2-orientation α and a flow ψ 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 (α,ψ) 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 bound is tight. In a similar way, 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.

Keywords

Graph representations (geometric and intersection representations, etc.), Density (toughness, etc.), square grid, Paths and cycles, interiorly disjoint grid paths, Planar graphs; geometric and topological aspects of graph theory

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
5
Average
Average
Average
gold