Drawings of Complete Multipartite Graphs up to Triangle Flips
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F23%3A10469712" target="_blank" >RIV/00216208:11320/23:10469712 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.4230/LIPIcs.SoCG.2023.6" target="_blank" >https://doi.org/10.4230/LIPIcs.SoCG.2023.6</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.4230/LIPIcs.SoCG.2023.6" target="_blank" >10.4230/LIPIcs.SoCG.2023.6</a>
Alternative languages
Result language
angličtina
Original language name
Drawings of Complete Multipartite Graphs up to Triangle Flips
Original language description
For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan's Theorem states that for any two simple drawings of the complete graph Kn with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (a.k.a. Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on n vertices is bounded by O(n^16). The latter proof uses a Carathéodory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the following sense: For the complete bipartite graph Km, n minus two edges and Km, n plus one edge for any m, n >= 4, as well as Kn minus a 4-cycle for any n >= 5, there exist two simple drawings with the same ERS that cannot be transformed into each other using triangle flips. So having the same ERS does not remain sufficient when removing or adding very few edges.
Czech name
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Czech description
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Classification
Type
D - Article in proceedings
CEP classification
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OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
Project
<a href="/en/project/GA22-19073S" target="_blank" >GA22-19073S: Combinatorial and computational complexity in topology and geometry</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Article name in the collection
Leibniz International Proceedings in Informatics, LIPIcs
ISBN
978-3-95977-273-0
ISSN
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e-ISSN
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Number of pages
16
Pages from-to
1-16
Publisher name
Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik
Place of publication
Dagstuhl
Event location
Dallas
Event date
Jun 12, 2023
Type of event by nationality
WRD - Celosvětová akce
UT code for WoS article
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