Orthogeodesic point-set embedding of trees

Authors:
Fabrizio Frati;Radoslav Fulek;Marcus Krug
Affiliations:
School of Basic Sciences, École Polytechnique Fédérale de Lausanne, Switzerland;School of Basic Sciences, École Polytechnique Fédérale de Lausanne, Switzerland;Institute of Theoretical Informatics, Karlsruhe Institute of Technology, Germany
Venue:
GD'11 Proceedings of the 19th international conference on Graph Drawing
Year:
2011

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Abstract

Let S be a set of N grid points in the plane, and let G a graph with n vertices (n≤N). An orthogeodesic point-set embedding of G on S is a drawing of G such that each vertex is drawn as a point of S and each edge is an orthogonal chain with bends on grid points whose length is equal to the Manhattan distance. We study the following problem. Given a family of trees $\mathcal F$ what is the minimum value f(n) such that every n-vertex tree in $\mathcal F$ admits an orthogeodesic point-set embedding on every grid-point set of size f(n)? We provide polynomial upper bounds on f(n) for both planar and non-planar orthogeodesic point-set embeddings as well as for the case when edges are required to be L-shaped chains.