Shuffles and Concatenations in Constructing of Graphs

TL;DR

This paper explores how shuffling and concatenation operations influence graph drawing, providing a syntactic framework for finite partial orders and explaining polyhedral collapses related to these operations.

Contribution

It introduces a complete syntactic description of shuffling and concatenation in finite partial orders and explains polyhedral collapses in the context of graph drawing.

Findings

Complete syntactic description of shuffling and concatenation operations.

Explanation of polyhedral collapses into associahedron, cyclohedron, and other polyhedra.

Connection of these polyhedra to categorial coherence questions.

Abstract

This is an investigation of the role of shuffling and concatenating in the theory of graph drawing. A simple syntactic description of these and related operations is proved complete in the context of finite partial orders, as general as possible. An explanation based on that is given for a previously investigated collapse of the permutohedron into the associahedron, and for collapses into other less familiar polyhedra, including the cyclohedron. Such polyhedra have been considered recently in connection with the notion of tubing, which is closely related to tree-like finite partial orders defined simply and investigated here in detail. Like the associahedron, some of these other polyhedra are involved in categorial coherence questions, which will be treated in a sequel to this paper. (Printed versions of our papers have sometimes been spoiled by referees and editorial assistants, and the versions posted here in the arXiv are always to be preferred. However, due to the irresponsible refereeing and the massive and misguided editorial assistance commissioned by Cambridge University Press, the text of the present paper printed in the journal "Mathematical Structures in Computer Science", where even the title has been changed without our consent, is such that we can bear responsibility only for the text posted here in the arXiv.)