Bar k-Visibility Graphs

Dean, Alice M.; Evans, William S.; Gethner, Ellen; Laison, Joshua D.; Safari, Mohammad Ali; Trotter, William T.

doi:10.7155/jgaa.00136

Cited by 51 publications

(70 citation statements)

References 15 publications

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“…A bar k-visibility representation is a bar visibility representation where each visibility can intersect at most k bars [50]. In other words, each edge can cross at most k vertices.…”

Section: Bar 1-visibility Representationsmentioning

confidence: 99%

An annotated bibliography on 1-planarity

Kobourov

Liotta

Montecchiani

2017

Computer Science Review

115

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The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at most another edge. The study of 1-planar graphs dates back to more than fifty years ago and, recently, it has driven increasing attention in the areas of graph theory, graph algorithms, graph drawing, and computational geometry. This annotated bibliography aims to provide a guiding reference to researchers who want to have an overview of the large body of literature about 1-planar graphs. It reviews the current literature covering various research streams about 1-planarity, such as characterization and recognition, combinatorial properties, and geometric representations. As an additional contribution, we offer a list of open problems on 1-planar graphs.

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“…A bar k-visibility representation is a bar visibility representation where each visibility can intersect at most k bars [50]. In other words, each edge can cross at most k vertices.…”

Section: Bar 1-visibility Representationsmentioning

confidence: 99%

An annotated bibliography on 1-planarity

Kobourov

Liotta

Montecchiani

2017

Computer Science Review

115

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“…This is witnessed by a point set S consisting of k points of the form {(i, 2i) | 0 ≤ i ≤ k − 1} and k − 1 points of the form {(2k − i, 2i − 1) | 1 ≤ i ≤ k − 1}, see Figure 4 for an example. It is easy to see that every pair of points in such a point set is in a common bottomless rectangle of size at most k. Finally, let us remark that an upper bound on c(k) for dynamic point sets in which points can both appear and disappear, as in Section 2, can be obtained by bounding the chromatic number of the corresponding so-called bar k-visibility graph, as defined by Dean et al [6]. In particular, they show that those graphs have O(kn) edges, yielding c(k) = O(k) for that case.…”

Section: Increasing the Number Of Colorsmentioning

confidence: 99%

Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles

Asinowski

Cardinal

Cohen

et al. 2013

Lecture Notes in Computer Science

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Abstract. We consider a coloring problem on dynamic, one-dimensional point sets: points appearing and disappearing on a line at given times. We wish to color them with k colors so that at any time, any sequence of p(k) consecutive points, for some function p, contains at least one point of each color.We prove that no such function p(k) exists in general. However, in the restricted case in which points appear gradually, but never disappear, we give a coloring algorithm guaranteeing the property at any time with p(k) = 3k − 2.This can be interpreted as coloring point sets in R 2 with k colors such that any bottomless rectangle containing at least 3k − 2 points contains at least one point of each color. Here a bottomless rectangle is an axis-aligned rectangle whose bottom edge is below the lowest point of the set.For this problem, we also prove a lower bound p(k) > ck, where c > 1.67. Hence for every k there exists a point set, every k-coloring of which is such that there exists a bottomless rectangle containing ck points and missing at least one of the k colors. Chen et al. (2009) proved that no such function p(k) exists in the case of general axis-aligned rectangles. Our result also complements recent results from Keszegh and Pálvölgyi on cover-decomposability of octants (2011, 2012).

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“…What is their book-thickness [15]? How are they related to classes of graphs that are defined by visibility representations [29,36]? There are partial answers for 1-planar graphs [5,6,18,23] for these problems, which need a generalization to other classes of beyond-planar graphs.…”

Section: Resultsmentioning

confidence: 99%

On fan-crossing and fan-crossing free graphs

Brandenburg

2018

Information Processing Letters

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Bar k-Visibility Graphs

Cited by 51 publications

References 15 publications

An annotated bibliography on 1-planarity

An annotated bibliography on 1-planarity

Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles

On fan-crossing and fan-crossing free graphs

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