TY - GEN
T1 - Drawings of Complete Multipartite Graphs up to Triangle Flips
AU - Aichholzer, Oswin
AU - Chiu, Man Kwun
AU - Hoang, Hung P.
AU - Hoffmann, Michael
AU - Kynčl, Jan
AU - Maus, Yannic
AU - Vogtenhuber, Birgit
AU - Weinberger, Alexandra
N1 - Publisher Copyright:
© Oswin Aichholzer, Man-Kwun Chiu, Hung P. Hoang, Michael Hoffmann, Jan Kynčl, Yannic Maus, Birgit Vogtenhuber, and Alexandra Weinberger; licensed under Creative Commons License CC-BY 4.0.
PY - 2023/6/1
Y1 - 2023/6/1
N2 - 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(n16). 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.
AB - 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(n16). 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.
KW - bipartite graphs
KW - complete graphs
KW - Gioan's Theorem
KW - k-partite graphs
KW - multipartite graphs
KW - Reidemeister moves
KW - Simple drawings
KW - simple topological graphs
KW - triangle flips
UR - http://www.scopus.com/inward/record.url?scp=85163474881&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SoCG.2023.6
DO - 10.4230/LIPIcs.SoCG.2023.6
M3 - Conference paper
AN - SCOPUS:85163474881
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 39th International Symposium on Computational Geometry, SoCG 2023
A2 - Chambers, Erin W.
A2 - Gudmundsson, Joachim
PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik
T2 - 39th International Symposium on Computational Geometry
Y2 - 12 June 2023 through 15 June 2023
ER -