Related papers: Higher-Order Triangular-Distance Delaunay Graphs: …

We investigate algorithms with predictions in computational geometry, specifically focusing on the basic problem of computing 2D Delaunay triangulations. Given a set $P$ of $n$ points in the plane and a triangulation $G$ that serves as a…

Strata of translation surfaces are covered by the closures of finitely many iso-Delaunay regions: open subspaces parametrizing surfaces whose Delaunay triangulations are combinatorially equivalent. We prove that the iso-Delaunay regions for…

Let $G$ be a graph each edge $e$ of which is given a length $\ell(e)$. This naturally induces a distance $d_\ell(x,y)$ between any two vertices $x,y$, and we let $\ell-TOP$ denote the completion of the corresponding metric space. It turns…

Given a point set $P$ and a class $\mathcal{C}$ of geometric objects, $G_\mathcal{C}(P)$ is a geometric graph with vertex set $P$ such that any two vertices $p$ and $q$ are adjacent if and only if there is some $C \in \mathcal{C}$…

Let $X$ be a finite, simple graph with vertex set $V(X)$. The $2$-distance graph $T_2(X)$ of $X$ is the graph with the same vertex set as $X$ and two vertices are adjacent if and only if their distance in $X$ is exactly $2$. A graph $G$ is…

Given two distinct point sets $P$ and $Q$ in the plane, we say that $Q$ \emph{blocks} $P$ if no two points of $P$ are adjacent in any Delaunay triangulation of $P\cup Q$. Aichholzer et al. (2013) showed that any set $P$ of $n$ points in…

We investigate structural and algorithmic advantages of a directed version of the well-researched class of distance-hereditary graphs. Since the previously defined distance-hereditary digraphs do not permit a recursive structure, we define…

We consider the flip-width of geometric graphs, a notion of graph width recently introduced by Toru\'nczyk. We prove that many different types of geometric graphs have unbounded flip-width. These include interval graphs, permutation graphs,…

Given a finite set $P\subset\mathbb{R}^2$, the directed Theta-6 graph, denoted $\vec{\Theta}_6(P)$, is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The $\vec{\Theta}_6(P)$-graph is defined as…

High triangle density -- the graph property stating that a constant fraction of two-hop paths belong to a triangle -- is a common signature of social networks. This paper studies triangle-dense graphs from a structural perspective. We prove…

Topological drawings are representations of graphs in the plane, where vertices are represented by points, and edges by simple curves connecting the points. A drawing is simple if two edges intersect at most in a single point, either at a…

We describe an algorithm that takes as input n points in the plane and a parameter {\epsilon}, and produces as output an embedded planar graph having the given points as a subset of its vertices in which the graph distances are a (1 +…

Delaunay flip is an elegant, simple tool to convert a triangulation of a point set to its Delaunay triangulation. The technique has been researched extensively for full dimensional triangulations of point sets. However, an important case of…

In [Distrance-regular Cayley graphs on dihedral groups, J. Combin. Theory Ser B 97 (2007) 14--33], Miklavi\v{c} and Poto\v{c}nik proposed the problem of characterizing distance-regular Cayley graphs, which can be viewed as an extension of…

The flip graph for a set $P$ of points in the plane has a vertex for every triangulation of $P$, and an edge when two triangulations differ by one flip that replaces one triangulation edge by another. The flip graph is known to have some…

The topic of this paper is related to the well-known notion of unit distance graphs. Take a graph with its edges coloured red and blue such that for some $d$ it can be mapped into the plane with all vertices going to distinct points, the…

The characterization of distance-regular Cayley graphs originated from the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, a classification of distance-regular Cayley…

Delaunay and Gabriel graphs are widely studied geometric proximity structures. Motivated by applications in wireless routing, relaxed versions of these graphs known as \emph{Locally Delaunay Graphs} ($LDGs$) and \emph{Locally Gabriel…

Let $G$ be a simple connected simple graph of order $n$. The distance Laplacian matrix $D^{L}(G)$ is defined as $D^L(G)=Diag(Tr)-D(G)$, where $Diag(Tr)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of…

The kd-tree and Bounding Volume Hierarchy (BVH) are well-known data structures for computing ray-object intersections. Less known is the Constrained Convex Space Partitioning (CCSP), which partitions space and makes the geometric primitives…