Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs

Aichholzer, Oswin ; García, Alfredo (Universidad de Zaragoza) ; Tejel, Javier (Universidad de Zaragoza) ; Vogtenhuber, Birgit ; Weinberger, Alexandra

Resumen: Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). We introduce a special kind of simple drawings that we call generalized twisted drawings. A simple drawing is generalized twisted if there is a point O such that every ray emanating from O crosses every edge of the drawing at most once and there is a ray emanating from O which crosses every edge exactly once.
Via this new class of simple drawings, we show that every simple drawing of the complete graph with n vertices contains Ω(n^{1/2}) pairwise disjoint edges and a plane path of length Ω((log n)/(log log n)). Both results improve over previously known best lower bounds. On the way we show several structural results about and properties of generalized twisted drawings. We further present different characterizations of generalized twisted drawings, which might be of independent interest.

Idioma: Inglés

DOI: 10.4230/LIPIcs.SoCG.2022.5

Año: 2022

Publicado en: Leibniz international proceedings in informatics 224 (2022), 5 [18 pp.]
ISSN: 1868-8969

Factor impacto SCIMAGO: 0.832 - Software - Logic

Financiación: info:eu-repo/grantAgreement/ES/DGA/E41-17R
Financiación: info:eu-repo/grantAgreement/EC/H2020/734922/EU/Combinatorics of Networks and Computation/CONNECT
Financiación: info:eu-repo/grantAgreement/ES/MICIU-AEI/PID2019-104129GB-I00-AEI-10.13039-501100011033

Tipo y forma: Article (Published version)

Área (Departamento): Área Estadís. Investig. Opera. (Dpto. Métodos Estadísticos)