Related papers: Bounded Degree Planar Geometric Spanners

The diameter of an undirected or a directed graph is defined to be the maximum shortest path distance over all pairs of vertices in the graph. Given an undirected graph $G$, we examine the problem of assigning directions to each edge of $G$…

We show that Delaunay triangulations and compressed quadtrees are equivalent structures. More precisely, we give two algorithms: the first computes a compressed quadtree for a planar point set, given the Delaunay triangulation; the second…

We introduce and study level-planar straight-line drawings with a fixed number $\lambda$ of slopes. For proper level graphs, we give an $O(n \log^2 n / \log \log n)$-time algorithm that either finds such a drawing or determines that no such…

A geometric $t$-spanner for a set $S$ of $n$ point sites is an edge-weighted graph for which the (weighted) distance between any two sites $p,q \in S$ is at most $t$ times the original distance between $p$ and~$q$. We study geometric…

A graph is $k$-degenerate if every subgraph has minimum degree at most $k$. We provide lower bounds on the size of a maximum induced 2-degenerate subgraph in a triangle-free planar graph. We denote the size of a maximum induced 2-degenerate…

We consider upward-planar layered drawings of directed graphs, i.e., crossing-free drawings in which each edge is drawn as a y-monotone curve going upward from its tail to its head, and the y-coordinates of the vertices are integers. The…

We present a self-contained short proof of the seminal result of Dillencourt (SoCG 1987 and DCG 1990) that Delaunay triangulations, of planar point sets in general position, are 1-tough. An important implication of this result is that…

In this paper, we show that given a weighted, directed planar graph $G$, and any $\epsilon >0$, there exists a polynomial time and $O(n^{\frac{1}{2}+\epsilon})$ space algorithm that computes the shortest path between two fixed vertices in…

Many of the classic graph problems cannot be solved in the Massively Parallel Computation setting (MPC) with strongly sublinear space per machine and $o(\log n)$ rounds, unless the 1-vs-2 cycles conjecture is false. This is true even on…

In this paper, we study the \textsf{Planar Disjoint Paths} problem: Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal is to find a set $\mathcal P$ of $k$ pairwise…

We describe a data structure, a rectangular complex, that can be used to represent hyperconvex metric spaces that have the same topology (although not necessarily the same distance function) as subsets of the plane. We show how to use this…

We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set $P$ of points in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle $\triangledown$, and there is an…

We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on $2n$ vertices are bounded by…

This paper introduces a Delaunay triangulation algorithm based on the external incremental method. Unlike traditional random incremental methods, this approach uses convex hull and points as basic operational units instead of triangles.…

The \emph{Delaunay graph} of a point set $P \subseteq \mathbb{R}^2$ is the plane graph with the vertex-set $P$ and the edge-set that contains $\{p,p'\}$ if there exists a disc whose intersection with $P$ is exactly $\{p,p'\}$. Accordingly,…

Delaunay Triangulation(DT) is one of the important geometric problems that is used in various branches of knowledge such as computer vision, terrain modeling, spatial clustering and networking. Kinetic data structures have become very…

We obtain better algorithms for computing more balanced orientations and degree splits in LOCAL. Important to our result is a connection to the hypergraph sinkless orientation problem [BMNSU, SODA'25] We design an algorithm of complexity…

We present a simple randomized scheme for triangulating a set $P$ of $n$ points in the plane, and construct a kinetic data structure which maintains the triangulation as the points of $P$ move continuously along piecewise algebraic…

We address the following problem: Given a complete $k$-partite geometric graph $K$ whose vertex set is a set of $n$ points in $\mathbb{R}^d$, compute a spanner of $K$ that has a ``small'' stretch factor and ``few'' edges. We present two…

We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have been recently applied to obtain various properties of…