6 research outputs found
Drawings of Complete Multipartite Graphs up to Triangle Flips
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
Flip Graph Connectivity for Arrangements of Pseudolines and Pseudocircles
Flip graphs of combinatorial and geometric objects are at the heart of many
deep structural insights and connections between different branches of discrete
mathematics and computer science. They also provide a natural framework for the
study of reconfiguration problems. We study flip graphs of arrangements of
pseudolines and of arrangements of pseudocircles, which are combinatorial
generalizations of lines and circles, respectively. In both cases we consider
triangle flips as local transformation and prove conjectures regarding their
connectivity.
In the case of pseudolines we show that the connectivity of the flip
graph equals its minimum degree, which is exactly . For the proof we
introduce the class of shellable line arrangements, which serve as reference
objects for the construction of disjoint paths. In fact, shellable arrangements
are elements of a flip graph of line arrangements which are vertices of a
polytope (Felsner and Ziegler; DM 241 (2001), 301--312). This polytope forms a
cluster of good connectivity in the flip graph of pseudolines. In the case of
pseudocircles we show that triangle flips induce a connected flip graph on
\emph{intersecting} arrangements and also on cylindrical intersecting
arrangements. The result for cylindrical arrangements is used in the proof for
intersecting arrangements. We also show that in both settings the diameter of
the flip graph is in . Our constructions make essential use of
variants of the sweeping lemma for pseudocircle arrangements (Snoeyink and
Hershberger; Proc.\ SoCG 1989: 354--363). We finally study cylindrical
arrangements in their own right and provide new combinatorial characterizations
of this class
Recommended from our members
Discrete Geometry (hybrid meeting)
A number of important recent developments in various branches of
discrete geometry were presented at the workshop, which took place in
hybrid format due to a pandemic situation. The presentations
illustrated both the diversity of the area and its strong connections
to other fields of mathematics such as topology, combinatorics,
algebraic geometry or functional analysis. The open questions abound
and many of the results presented were obtained by young researchers,
confirming the great vitality of discrete geometry
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Arrangements of Approaching Pseudo-Lines
We consider arrangements of n pseudo-lines in the Euclidean plane where each pseudo-line ℓi is represented by a bi-infinite connected x-monotone curve fi(x), x∈R, such that for any two pseudo-lines ℓi and ℓj with i<j, the function x↦fj(x)−fi(x) is monotonically decreasing and surjective (i.e., the pseudo-lines approach each other until they cross, and then move away from each other). We show that such arrangements of approaching pseudo-lines, under some aspects, behave similar to arrangements of lines, while for other aspects, they share the freedom of general pseudo-line arrangements. For the former, we prove:
There are arrangements of pseudo-lines that are not realizable with approaching pseudo-lines.
Every arrangement of approaching pseudo-lines has a dual generalized configuration of points with an underlying arrangement of approaching pseudo-lines.
For the latter, we show:
There are 2Θ(n2) isomorphism classes of arrangements of approaching pseudo-lines (while there are only 2Θ(nlogn) isomorphism classes of line arrangements).
It can be decided in polynomial time whether an allowable sequence is realizable by an arrangement of approaching pseudo-lines.
Furthermore, arrangements of approaching pseudo-lines can be transformed into each other by flipping triangular cells, i.e., they have a connected flip graph, and every bichromatic arrangement of this type contains a bichromatic triangular cell.ISSN:0179-5376ISSN:1432-044