Fabrizio Frati
The flip graph is the graph whose nodes correspond to non-isomorphic combinatorial triangulations and whose edges connect pairs of triangulations that can be obtained one from the other by flipping a single edge. In this note we show that the diameter of the flip graph is at least $\frac{7n}{3} + Θ(1)$, improving upon the previous $2n + Θ(1)$ lower bound.
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The Flip Diameter of Rectangulations and Convex Subdivisions
研究平面上$n$点矩形剖分与凸剖分的构型空间，证明$O(n\log n)$次翻转和旋转操作可变换任意两个矩形剖分，且某些点集上该界最优，部分结论推广至凸剖分。
Flips in Odd Matchings
本研究针对平面一般位置的奇数个点集，定义了含m条边的平面匹配构成的图，证明其连通性并给出直径O(n²)的上界，同时得到凸位置点翻转图直径为Θ(n)的紧界。
Flip Distance of Non-Crossing Spanning Trees: NP-Hardness and Improved Bounds
研究非交叉生成树翻转距离的NP难性及改进翻转图直径的界限，通过冲突图工具证明计算翻转距离为NP难问题，并改进翻转图直径下界。
Flips in triangulations have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the flipping process has not been addressed. We examine this question by attaching unique labels to the triangulation edges. We introduce the concept of the orbit of an edge $e$, which is the set of all edges reachable from $e$ via flips. We establish the first upper and lower bounds on the diameter of the flip graph in this setting. Specifically, we prove tight $Θ(n \log n)$ bounds for edge-labelled triangulations of $n$-vertex convex polygons and c…
Constrained Flips in Plane Spanning Trees
本文研究凸位置点集上平面生成树的兼容翻转与旋转两类受限翻转，将两种翻转图直径的上界分别改进为 $\frac{5n}{3}-O(1)$ 和 $\frac{7n}{4}-O(1)$，并利用“快乐边”性质给出计算最短兼容翻转序列的固定参数可解算法。
In this paper we consider the flip operation for combinatorial pointed pseudo-triangulations where faces have size 3 or 4, so-called combinatorial 4-PPTs. We show that every combinatorial 4-PPT is stretchable to a geometric pseudo-triangulation, which in general is not the case if faces may have size larger than 4. Moreover, we prove that the flip graph of combinatorial 4-PPTs is connected and has diameter $O(n^2)$, even in the case of labeled vertices with fixed outer face. For this case we provide an $Ω(n\log n)$ lower bound.
A realizer, commonly known as Schnyder woods, of a triangulation is a partition of its interior edges into three oriented rooted trees. A flip in a realizer is a local operation that transforms one realizer into another. Two types of flips in a realizer have been introduced: colored flips and cycle flips. A corresponding flip graph is defined for each of these two types of flips. The vertex sets are the realizers, and two realizers are adjacent if they can be transformed into each other by one flip. In this paper we study the relation between these two types of flips and their corresponding fl…
We show that any combinatorial triangulation on n vertices can be transformed into a 4-connected one using at most floor((3n - 9)/5) edge flips. We also give an example of an infinite family of triangulations that requires this many flips to be made 4-connected, showing that our bound is tight. In addition, for n >= 19, we improve the upper bound on the number of flips required to transform any 4-connected triangulation into the canonical triangulation (the triangulation with two dominant vertices), matching the known lower bound of 2n - 15. Our results imply a new upper bound on the diameter …
A Near-Optimal Space Lower Bound for Euclidean Diameter Estimation in Dynamic Streams
针对动态流中欧几里得直径估计，我们证明了任何c近似算法需$n^{\tilde{\Omega}(1/c^2)}$空间，与Indyk的上界匹配，解决了空间复杂度问题。
We show that every triangulation (maximal planar graph) on $n\ge 6$ vertices can be flipped into a Hamiltonian triangulation using a sequence of less than $n/2$ combinatorial edge flips. The previously best upper bound uses $4$-connectivity as a means to establish Hamiltonicity. But in general about $3n/5$ flips are necessary to reach a $4$-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on $n$ vertices from $5.2n-33.6$ to $5n-23$. We also show that for every triangulation on $n$ vertices there is a simultaneous fli…
In this paper we investigate the structure of flip graphs on non-crossing perfect matchings in the plane. Specifically, consider all non-crossing straight-line perfect matchings on a set of $2n$ points that are placed equidistantly on the unit circle. A flip operation on such a matching replaces two matching edges that span an empty quadrilateral with the other two edges of the quadrilateral, and the flip is called centered if the quadrilateral contains the center of the unit circle. The graph $\mathcal{G}_n$ has those matchings as vertices, and an edge between any two matchings that differ in…
On the Connectivity of the Flip Graph of Plane Spanning Paths
针对平面生成路径翻转图连通性这一开放问题，论文研究固定端点与后缀无关路径等诱导子图，给出直径和半径紧界，并证明至多两个凸层点集上的连通性。
Flip Graph Connectivity for Arrangements of Pseudolines and Pseudocircles
本文研究伪线和伪圆排列的翻转图连通性，证明伪线翻转图连通性等于最小度n-2，伪圆翻转图在相交排列及柱面相交排列上连通，并证明两类翻转图直径均为Θ(n^3)。
The Origami flip graph of the $2\times n$ Miura-ori
研究 Miura-ori 折痕图案的局部有效山谷-山峰赋值，通过枚举 $2\times n$ 情形的翻转图顶点和边，证明顶点度数的多项式描述，并确定其直径为 $\lceil n^2/2 \rceil$。
Flipping odd matchings in geometric and combinatorial settings
研究奇匹配的重配置问题，提出组合场景中任意两奇匹配可相互重配置的图的多项式时间特征，证明相关翻转图直径线性及转换问题NP完全，且凸点集几何场景中该问题为FPT。
Flipping Non-Crossing Spanning Trees
该研究聚焦平面点集非交叉生成树翻转图的直径问题，通过引入冲突图最大无环子集转化方法，给出凸位置下直径的新上下界，推翻了原有3n/2上界猜想，为重构问题研究提供了新工具。
A Polynomial Space Lower Bound for Diameter Estimation in Dynamic Streams
该研究针对动态流模型下的直径估计问题，证明常数因子近似需多项式空间，建立了动态流算法与尺度不变函数线性概要、图最小秩的关联，并给出近乎匹配的上界结果。
Maximal origami flip graphs of flat-foldable vertices: properties and algorithms
该研究首次探究单顶点折痕模式的有效MV赋值通过面翻转的关联，证明等角单顶点折痕模式A₂ₙ的折纸翻转图是同度数中最大的，且连通、直径为n，给出两种O(n²)遍历算法并枚举其图参数。
Charting the Diameter Computation Landscape of Geometric Intersection Graphs in Three Dimensions and Higher
本文研究了三维及更高维度几何相交图直径计算问题，提出首个三维对象直径3的亚二次时间算法，并给出单位球体在直径3下的亚二次时间下界，同时扩展了矩形框直径2和3的算法与下界。
Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation is a full triangulation of some subset P' of P containing all extreme points in P. A bistellar flip on a partial triangulation flips an edge (an edge flip), removes a non-extreme point of degree 3, or adds a point in P \ P' as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The edge flip…
For a set $P$ of $n$ points in the plane in general position, a non-crossing spanning tree is a spanning tree of the points where every edge is a straight-line segment between a pair of points and no two edges intersect except at a common endpoint. We study the problem of reconfiguring one non-crossing spanning tree of $P$ to another using a sequence of flips where each flip removes one edge and adds one new edge so that the result is again a non-crossing spanning tree of $P$. There is a known upper bound of $2n-4$ flips [Avis and Fukuda, 1996] and a lower bound of $1.5n - 5$ flips. We give a …
ETH Flippers Approach to Parallel Reconfiguration of Triangulations: SAT formulation and Heuristics
本文提出了一种结合精确求解器和启发式方法的算法，用于在三角剖分中寻找最小并行翻转距离的中心三角剖分，解决了CG:SHOP 2026挑战中的问题。
Reconfiguration of plane trees in convex geometric graphs
本文研究凸几何图中非交叉生成树之间的翻转重构问题，首次以线性因子将翻转序列长度上界改进为约1.95n，并将下界改进为对称差的5/3倍，同时推广到非交叉翻转和旋转操作。
We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called \emph{pseudo-simultaneously flippable edges}. Such edges are related to the notion of convex decompositions spanned by S. We prove a worst-case tight lower bound for the number of pseudo-simultaneously flippable edges in a triangulation in terms of the number of vertices. We use this bound for deriving new upper bounds for the maximal number of crossing-free straight-edge graphs that can be embedded on any fixed set of N points in the plane. We obtain new …
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