Combinatorial Characterization of Upward Planarity

Lu, Xuexing; Ye, Yu

doi:10.1007/s40304-018-0169-2

Commun. Math. Stat.

2019

DOI: 10.1007/s40304-018-0169-2

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Combinatorial Characterization of Upward Planarity

Xuexing Lu

Yu Ye

Abstract: We give a combinatorial characterization of upward planar graphs in terms of upward planar orders, which are special linear extensions of edge posets.

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“…Similar to the combinatorial characterization of a PPG-class, there is a totally combinatorial characterization of a progressive plane graph (without isolated vertices) by the notion of a UPO-graph (abbreviation of upward planarly ordered graph)[13].…”

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confidence: 99%

A Graphical Calculus for Semi-Groupal Categories

2018

Appl Categor Struct

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Around the year 1988, Joyal and Street established a graphical calculus for monoidal categories, which provides a firm foundation for many explorations of graphical notations in mathematics and physics. For a deeper understanding of their work, we consider a similar graphical calculus for semi-groupal categories. We introduce two frameworks to formalize this graphical calculus, a topological one based on the notion of a processive plane graph and a combinatorial one based on the notion of a planarly ordered processive graph, which serves as a combinatorial counterpart of a deformation class of processive plane graphs. We demonstrate the equivalence of Joyal and Street's graphical calculus and the theory of upward planar drawings. We introduce the category of semi-tensor schemes, and give a construction of a free monoidal category on a semi-tensor scheme. We deduce the unit convention as a kind of quotient construction, and show an idea to generalize the unit convention. Finally, we clarify the relation of the unit convention and Joyal and Street's construction of a free monoidal category on a tensor scheme.

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A Graphical Calculus for Semi-Groupal Categories

2018

Appl Categor Struct

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Combinatorial Characterization of Upward Planarity

Abstract: We give a combinatorial characterization of upward planar graphs in terms of upward planar orders, which are special linear extensions of edge posets.

Cited by 1 publication

References 9 publications

A Graphical Calculus for Semi-Groupal Categories

A Graphical Calculus for Semi-Groupal Categories

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