Natural Sciences and Engineering Research Council of Canada
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Monotone Simultaneous Paths Embeddings in $\mathbb{R}^d$
David Bremner (1), Olivier Devillers (2), Marc Glisse (3), Sylvain Lazard (2), Giuseppe Liotta (4), Tamara Mchedlidze (5), Guillaume Moroz (2), Sue Whitesides (6), Stephen Wismath (7)
(1) Faculty of Computer Science
(2) Geometric Algorithms and Models Beyond the Linear and Euclidean realm
(3) Understanding the Shape of Data
(4) Dipartimento di Matematica e Informatica [Perugia]
(5) Institute of Theoretical Informatics
(6) Department of Computer Science [Victoria]
(7) Department of Mathematics and Computer Science
Abstract
We study the following problem: Given $k$ paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that for any dimension $d\geq 2$, there is a set of $d + 1$ paths that does not admit a monotone simultaneous geometric embedding.
