August 20, 2020
An interesting class of orthogonal representations consists of the so-called turn-regular ones, i.e., those that do not contain any pair of reflex corners that "point to each other" inside a face. For such a representation H it is possible to compute in linear time a minimum-area drawing, i.e., a drawing of minimum area over all possible assignments of vertex and bend coordinates of H. In contrast, finding a minimum-area drawing of H is NP-hard if H is non-turn-regular. This ...
February 5, 2021
The Minimum Linear Arrangement problem (MLA) consists of finding a mapping $\pi$ from vertices of a graph to distinct integers that minimizes $\sum_{\{u,v\}\in E}|\pi(u) - \pi(v)|$. In that setting, vertices are often assumed to lie on a horizontal line and edges are drawn as semicircles above said line. For trees, various algorithms are available to solve the problem in polynomial time in $n=|V|$. There exist variants of the MLA in which the arrangements are constrained. Ior...
September 5, 2017
Let $T$ be an $n$-node tree of maximum degree 4, and let $P$ be a set of $n$ points in the plane with no two points on the same horizontal or vertical line. It is an open question whether $T$ always has a planar drawing on $P$ such that each edge is drawn as an orthogonal path with one bend (an "L-shaped" edge). By giving new methods for drawing trees, we improve the bounds on the size of the point set $P$ for which such drawings are possible to: $O(n^{1.55})$ for maximum deg...
January 2, 2021
Let $\mathcal{P}$ be a finite set of points in the plane in general position. For any spanning tree $T$ on $\mathcal{P}$, we denote by $|T|$ the Euclidean length of $T$. Let $T_{\text{OPT}}$ be a plane (that is, noncrossing) spanning tree of maximum length for $\mathcal{P}$. It is not known whether such a tree can be found in polynomial time. Past research has focused on designing polynomial time approximation algorithms, using low diameter trees. In this work we initiate a s...
October 25, 2019
A planar orthogonal drawing $\Gamma$ of a planar graph $G$ is a geometric representation of $G$ such that the vertices are drawn as distinct points of the plane, the edges are drawn as chains of horizontal and vertical segments, and no two edges intersect except at their common end-points. A bend of $\Gamma$ is a point of an edge where a horizontal and a vertical segment meet. $\Gamma$ is bend-minimum if it has the minimum number of bends over all possible planar orthogonal d...
June 7, 2016
In this paper, we study how to draw trees so that they are planar, straight-line and respect a given order of edges around each node. We focus on minimizing the height, and show that we can always achieve a height of at most 2pw(T)+1, where pw(T) (the so-called pathwidth) is a known lower bound on the height. Hence we give an asymptotic 2-approximation algorithm. We also create a drawing whose height is at most 3pw(T ), but where the width can be bounded by the number of node...
August 31, 2010
We show that every graph of maximum degree three can be drawn in three dimensions with at most two bends per edge, and with 120-degree angles between any two edge segments meeting at a vertex or a bend. We show that every graph of maximum degree four can be drawn in three dimensions with at most three bends per edge, and with 109.5-degree angles, i.e., the angular resolution of the diamond lattice, between any two edge segments meeting at a vertex or bend.
March 5, 2017
We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let $n$ denote the number of vertices of a graph. We show that trees can be drawn with $3n/4$ straight-line segments on a polynomial grid, and with $n/2$ straight-line segments on a quasi-poly...
August 12, 2014
We tackle the problem of constructing increasing-chord graphs spanning point sets. We prove that, for every point set P with n points, there exists an increasing-chord planar graph with O(n) Steiner points spanning P. Further, we prove that, for every convex point set P with n points, there exists an increasing-chord graph with O(n log n) edges (and with no Steiner points) spanning P.
March 20, 2019
The visual complexity of a graph drawing can be measured by the number of geometric objects used for the representation of its elements. In this paper, we study planar graph drawings where edges are represented by few segments. In such a drawing, one segment may represent multiple edges forming a path. Drawings of planar graphs with few segments were intensively studied in the past years. However, the area requirements were only considered for limited subclasses of planar gra...