Bimonotone Subdivisions of Point Configurations in the Plane

Word-representability of subdivisions of triangular grid graphs

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $(x,y)\in E$. A triangular grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A subdivision of a triangular grid graph is replacing some of its cells by plane copies of the complete graph $K_4$. Inspired by a recent elegant result of Akrobotu et al., who classified word-representable triangulations of grid graphs related to convex polyominoes, we characterize word-r…

Real Tropical Singularities and Bergman Fans

In this paper, we classify singular real plane tropical curves by means of subdivisions of Newton polytopes. First, we introduce signed Bergman fans (generalizing positive Bergman fans from [AKW06]) that describe real tropicalizations of real linear spaces ([Tab15]). Then, we establish a duality of real plane tropical curves and signed regular subdivisions of the Newton polytope and explore the combinatorics. We define a signed secondary fan that parametrizes real tropical Laurent polynomials and study the subset providing singular real plane tropical curves. A cone of the signed secondary fan…

Point-Line Geometry in the Tropical Plane

We study the classical result by Bruijn and Erd\H os regarding the bound on the number of lines determined by a $n$-point configuration in the plane, and in the light of the recently proven Tropical Sylvester-Gallai theorem, come up with a tropical version of the above-mentioned result. In this work, we introduce stable tropical lines, which help in answering questions pertaining to incidence geometry in the tropical plane. Projective duality in the tropical plane helps in translating the question for stable lines to stable intersections that have been previously studied in depth. Invoking dua…

Intersection Graphs of L-Shapes and Segments in the Plane

An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: L, Γ, LE{} and \eeG. A $k$-bend path is a simple path in the plane, whose direction changes $k$ times from horizontal to vertical. If a graph admits an intersection representation in which every vertex is represented by an L, an L{} or Γ, a $k$-bend path, or a segment, then this graph is called an $\{L\}$-graph, $\{L,Γ\}$-graph, $B_k$-VPG-graph or SEG-graph, respectively. Motivated by a theorem of Middendorf and Pfeiffer [Discrete Mathematics, 108(1):365--372, 1992], stating tha…

矩形剖分与凸剖分的翻转直径

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。

带关联几何应用的二部细分图的非平衡Zarankiewicz问题

Unbalanced Zarankiewicz problem for bipartite subdivisions with applications to incidence geometry

该研究确定了完全二部细分图$K_{s,t}'$的线性阈值上界及下界条件，并将其应用于复平面点线关联数估计与平面不同距离数下界推导，推进了极值图论与关联几何的交叉研究。