Grid Drawing Research Articles

Convex grid drawings of planar graphs with constant edge-vertex resolution

In this work, we continue the study of the area required for convex straight-line grid drawings of 3-connected plane graphs, which has been intensively investigated in the last decades. Motivated by applications, such as graph editors, we additionally require the obtained drawings to have bounded edge-vertex resolution, that is, the closest distance between a vertex and any non-incident edge in the drawing is lower bounded by a constant that does not depend on the size of the graph. We present a drawing algorithm that takes as input a 3-connected plane graph with n vertices and f internal faces, and computes a convex straight-line drawing with edge-vertex resolution at least 12 on an integer grid of size (n−2+a)×(n−2+a), where a=min⁡{n−3,f}. Our result improves the previously best-known area bound of (3n−7)×(3n−7)/2 by Chrobak, Goodrich and Tamassia.

Convex Grid Drawings of Internally Triconnected Plane Graphs with Pentagonal Contours

Convex Grid Drawings of Internally Triconnected Plane Graphs with Pentagonal Contours

Morphing tree drawings in a small 3D grid

We study crossing-free grid morphs for planar drawings of trees using the third dimension. A morph consists of morphing steps, where vertices of the tree move simultaneously along straight-line trajectories at constant speeds. The aim is to find a morph between two given drawings of the same tree that has a small number of steps and is crossing-free, i.e. no two vertices overlap or no two edges of the tree intersect except in their common endpoints at any moment during the morph. It is already known, that there exists a crossing-free morph between two drawings of an $n$-vertex planar graph $G$ with $\mathcal{O}(n)$ morphing steps, and that using the third dimension the number of steps can be reduced to $\mathcal{O}(\log n)$ for an $n$-vertex tree. However, these morphs do not bound one practical parameter, the resolution. Can the number of steps still be reduced substantially by using the third dimension with an additional restriction of keeping the resolution bounded throughout the morph? We answer this question in affirmative by presenting a $3D$ non-crossing morph between two planar grid drawings of an $n$-vertex tree in $\mathcal{O}(\sqrt{n} \log n)$ morphing steps such that each intermediate drawing is a grid drawing, i.e., vertices of the tree are mapped to the nodes of a $3D$ grid of polynomial volume.

Grid Drawings of Five-Connected Plane Graphs

Grid Drawings of Five-Connected Plane Graphs

Convex Grid Drawings of Plane Graphs with Pentagonal Contours on <i>O</i>(<i>n</i><sup>2</sup>) Grids

In a convex grid drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection, all vertices are put on grid points and all facial cycles are drawn as convex polygons. A plane graph G has a convex drawing if and only if G is internally triconnected, and an internally triconnected plane graph G has a convex grid drawing on an (n-1)×(n-1) grid if either G is triconnected or the triconnected component decomposition tree T(G) of G has two or three leaves, where n is the number of vertices in G. An internally triconnected plane graph G has a convex grid drawing on a 2n×2n grid if T(G) has exactly four leaves. Furthermore, an internally triconnected plane graph G has a convex grid drawing on a 6n×n2 grid if T(G) has exactly five leaves. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 20n×16n grid if T(G) has exactly five leaves. We also present an algorithm to find such a drawing in linear time. This is the first algorithm that finds a convex grid drawing of such a plane graph G in a grid of O(n2) size.

Grid drawings of graphs with constant edge-vertex resolution

We study the algorithmic problem of computing drawings of graphs in which (i) each vertex is a disk with fixed radius ρ, (ii) each edge is a straight-line segment connecting the centers of the two disks representing its end-vertices, (iii) no two disks intersect, and (iv) the distance between an edge segment and the center of a non-incident disk, called edge-vertex resolution, is at least ρ. We call such drawings disk-link drawings.In this paper we focus on the case of constant edge-vertex resolution, namely ρ=12 (i.e., disks of unit diameter). We prove that star graphs, which trivially admit straight-line drawings in linear area, require quadratic area in any such disk-link drawing. On the positive side, we present constructive techniques that yield improved upper bounds for the area requirements of disk-link drawings for several (planar and nonplanar) graph classes, including bounded bandwidth, complete, and planar graphs. In particular, the presented bounds for complete and planar graphs are asymptotically tight.

The Application of the Power Grid Map Intelligent Recognition Technology in the Power Grid Early Warning System

As the power grid enterprises grow, the relationship between their power grid systems is becoming closer and closer. In the context of industrial development with the energy structure unbalanced, The operation of the power grid becomes more and more complicated, making the power grid system face the technical problems. In this context of technological development, power grid enterprises should follow the pace of the times and integrate intelligent technology into the technological innovation of the power grid to make the ability to predict the power grid fault come true in advance, and the grid map intelligent identification technology is the key technology to support the power grid enterprise to realize informatization and intellectualization. Because most of the original data of the power grid enterprise are stored in various kinds of power grid design drawings, the power grid intelligent identification technology can effectively visualize the data of such power grid drawings, and then realize the intellectualization which early warning technology means.

Drawing a rooted tree as a rooted y−monotone minimum spanning tree

Given a set P of points in the plane, and a point r∈P identified as the root of P, the rooted y−Monotone Minimum Spanning Tree (rooted y−MMST) of P is the connected spanning geometric graph of P in which all the vertices are connected to the root by some y−monotone path and the sum of the Euclidean lengths of its edges is the minimum. We give a linear-time algorithm that draws any rooted tree as a rooted y−MMST. A corollary of the previous sentence is that for every natural number M there exists a rooted y−MMST of maximum degree M. We also show that there exist n-node rooted trees for which any grid drawing as a rooted y-MMST requires a grid of area exponential in n.

Drawing plane triangulations with few segments

Dujmović et al. (2007) [6] showed that every 3-connected plane graph G with n vertices admits a straight-line drawing with at most 2.5n−3 segments, which is also the best known upper bound when restricted to plane triangulations. On the other hand, they showed that there exist triangulations requiring 2n−6 segments. In this paper we show that every plane triangulation admits a straight-line drawing with at most (7n−2Δ0−10)/3≤2.33n segments, where Δ0 is the number of cyclic faces in the minimum realizer of G. For general plane graphs with n vertices and m edges, our algorithm requires at most (16n−3m−28)/3≤5.33n−m segments, which is less than 2.5n−3 for all m≥2.84n. In the context of grid drawings, where the vertices are restricted to have integer coordinates, we show that every triangulation with maximum degree D can be drawn with at most 2n+t−3 segments and O(8t⋅D2t) area, where t is the minimum number of leaves over all the trees of the minimum realizer. This is the first nontrivial attempt to simultaneously optimize the area and the number of segments while drawing triangulations. These results extend to the case when the goal is to optimize the number of slopes in the drawing.

Layered separators in minor-closed graph classes with applications

Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Ω(n) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minor-closed classes. We study a special type of graph separator, called a layered separator, which may have linear size in n, but has bounded size with respect to a different measure, called the width. We prove, for example, that planar graphs and graphs of bounded Euler genus admit layered separators of bounded width. More generally, we characterise the minor-closed classes that admit layered separators of bounded width as those that exclude a fixed apex graph as a minor.We use layered separators to prove O(log⁡n) bounds for a number of problems where O(n) was a long-standing previous best bound. This includes the nonrepetitive chromatic number and queue-number of graphs with bounded Euler genus. We extend these results with a O(log⁡n) bound on the nonrepetitive chromatic number of graphs excluding a fixed topological minor, and a logO(1)⁡n bound on the queue-number of graphs excluding a fixed minor. Only for planar graphs were logO(1)⁡n bounds previously known. Our results imply that every n-vertex graph excluding a fixed minor has a 3-dimensional grid drawing with nlogO(1)⁡n volume, whereas the previous best bound was O(n3/2).

Does listening to Mozart's music influence visuospatial short-term memory in young adults?

IntroductionMusic is claimed to improve mental function and many researchers claim that this effect related to Mozart's music is limited to enhancement of the spatial temporal reasoning and not to other cognitive functions.ObjectivesTo explore the influence of Mozart's music on visuospatial memory.MethodsSixty adults (37 women and 23 men), with Mage = 21.83, SDage = 2.38, Meducation = 14.03, SDeducation = .99, and without any formal musical education were examined through an experimental process. Participants in groups of ten listened to Mozart's sonata for two pianos in D major, K.448, to Mozart's violin concerto No.3 in G major, K.216, and to a no sounds condition in varying order. The participants after listening to each 10-minute condition were presented with a series of randomly generated patterns made up of black squares on a chess-like surface. This was used in order to test the storage capacity of their visuospatial memory. After 3 seconds of presentation for each drawing, they were asked to reproduce by drawing these patterns that progressively got bigger.ResultsResults revealed for all three conditions that the number of correct grid drawings made by the participants was not significantly statistically different (P > 05), and therefore their visuospatial memory retention was not influenced by any kind of music.ConclusionsFuture research could examine in more detail the retention and manipulation of visuospatial information not only in tasks similar to the visual patterns test, but also in different tests used for clinical and non-clinical populations.Disclosure of interestThe authors have not supplied their declaration of competing interest.

On some properties of doughnut graphs

The class of doughnut graphs is a subclass of 5-connected planar graphs. It is known that a doughnut graph admits a straight-line grid drawing with linear area, the outerplanarity of a doughnut graph is 3, and a doughnut graph is k-partitionable. In this paper we show that a doughnut graph exhibits a recursive structure. We also give an efficient algorithm for finding a shortest path between any pair of vertices in a doughnut graph. We also propose a nice application of a doughnut graph based on its properties.

Good spanning trees in graph drawing

A plane graph is a planar graph with a fixed planar embedding. In this paper we define a special spanning tree of a plane graph which we call a good spanning tree. Not every plane graph has a good spanning tree. We show that every connected planar graph has a planar embedding with a good spanning tree. Using a good spanning tree, we show that every connected planar graph G of n vertices has a straight-line monotone grid drawing on an O(n)×O(n2) grid, and such a drawing can be found in O(n) time. Our results solve two open problems on monotone drawings of planar graphs posed by Angelini et al. Using good spanning trees, we also give simple linear-time algorithms for finding a 2-visibility representation of a connected planar graph G of n vertices on a (2n−1)×(2n−1) grid and for finding a spike-VPG representation of G on a (2n−1)×n grid.

Straight-line monotone grid drawings of series–parallel graphs

A monotone drawing of a planar graph G is a planar straight-line drawing of G where a monotone path exists between every pair of vertices of G in some direction. Recently monotone drawings of graphs have been discovered as a new standard for visualizing graphs. In this paper we study monotone drawings of series–parallel graphs in a variable embedding setting. We show that a series–parallel graph of n vertices has a straight-line planar monotone drawing on a grid of size O(n) × O(n2) and such a drawing can be found in linear time.

Outer 1-Planar Graphs

A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are in the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time, and specialize 1-planar graphs, whose recognition is $${NP}$$NP-hard. We explore o1p graphs. Our first main result is a linear-time algorithm that takes a graph as input and returns a positive or a negative witness for o1p. If a graph $$G$$G is o1p, then the algorithm computes an embedding and can augment $$G$$G to a maximal o1p graph. Otherwise, $$G$$G includes one of six minors, which is detected by the recognition algorithm. Secondly, we establish structural properties of o1p graphs. o1p graphs are planar and are subgraphs of planar graphs with a Hamiltonian cycle. They are neither closed under edge contraction nor under subdivision. Several important graph parameters, such as treewidth, colorability, stack number, and queue number, increase by one from outerplanar to o1p graphs. Every o1p graph of size $$n$$n has at most $$\frac{5}{2} n - 4$$52n-4 edges and there are maximal o1p graphs with $$\frac{11}{5} n - \frac{18}{5}$$115n-185 edges, and these bounds are tight. Finally, every o1p graph has a straight-line grid drawing in $$\fancyscript{O}(n^2)$$O(n2) area with all vertices in the outer face, a planar visibility representation in $$\fancyscript{O}(n \log n)$$O(nlogn) area, and a 3D straight-line drawing in linear volume, and these drawings can be constructed in linear time.

A straight-line order-preserving binary tree drawing algorithm with linear area and arbitrary aspect ratio

Graph layouts and visualizations have been at the forefront of graph drawing research for decades, consequently leading to aesthetic heuristics that not only generate better visualizations and aesthetically appealing graphs but also improve readability and understanding of the graphs. A variety of approaches examines aesthetics of nodes, edges, or graph layout, and related readability metrics. Improving the known bounds on two aesthetic requirements (area and aspect ratio) for planar straight-line order-preserving grid drawings of binary trees is presented in an algorithm that uses a separation approach. The new bounds are optimal in area and aspect ratio, where the optimum values are linear and 1:1 respectively.

Graph layouts via layered separators

A k-queue layout of a graph consists of a total order of the vertices, and a partition of the edges into k sets such that no two edges that are in the same set are nested with respect to the vertex ordering. A k-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The queue-number (track-number) of a graph G, is the minimum k such that G has a k-queue (k-track) layout.This paper proves that every n-vertex planar graph has track number and queue number at most O(log⁡n). This improves the result of Di Battista, Frati, and Pach (2013) [5] who proved the first sub-polynomial bounds on the queue number and track number of planar graphs. Specifically, they obtained O(log2⁡n) queue number and O(log8⁡n) track number bounds for planar graphs.The result also implies that every planar graph has a 3D crossing-free grid drawing in O(nlog⁡n) volume. The proof uses a non-standard type of graph separators.

Crossings in Grid Drawings

We prove tight crossing number inequalities for geometric graphs whose vertex sets are taken from a $d$-dimensional grid of volume $N$ and give applications of these inequalities to counting the number of crossing-free geometric graphs that can be drawn on such grids.In particular, we show that any geometric graph with $m\geq 8N$ edges and with vertices on a 3D integer grid of volume $N$, has $\Omega((m^2/N)\log(m/N))$ crossings. In $d$-dimensions, with $d\ge 4$, this bound becomes $\Omega(m^2/N)$. We provide matching upper bounds for all $d$. Finally, for $d\ge 4$ the upper bound implies that the maximum number of crossing-free geometric graphs with vertices on some $d$-dimensional grid of volume $N$ is $N^{\Theta(N)}$. In 3 dimensions it remains open to improve the trivial bounds, namely, the $2^{\Omega(N)}$ lower bound and the $N^{O(N)}$ upper bound.

Convex Grid Drawings of Plane Graphs with Pentagonal Contours

Convex Grid Drawings of Plane Graphs with Pentagonal Contours

Reprint of: Grid representations and the chromatic number

A grid drawing of a graph maps vertices to the grid Zd and edges to line segments that avoid grid points representing other vertices. We show that a graph G is qd-colorable, d, q⩾2, if and only if there is a grid drawing of G in Zd in which no line segment intersects more than q grid points. This strengthens the result of D. Flores Pen̋aloza and F.J. Zaragoza Martinez. Second, we study grid drawings with a bounded number of columns, introducing some new NP-complete problems. Finally, we show that any planar graph has a planar grid drawing where every line segment contains exactly two grid points. This result proves conjectures asked by D. Flores Pen̋aloza and F.J. Zaragoza Martinez.