Alice M. Dean

Bar k-Visibility Graphs

Let S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a one-to-one correspondence between vertices of G and bars in S, such that there is an edge between two vertices in G if and only if there exists an unobstructed vertical line of sight between their corresponding bars. If bars are allowed to see through each other, the graphs representable in this way are precisely the interval graphs. We consider representations in which bars are allowed to see through at most k other bars. Since all bar visibility graphs are planar, we seek measurements of closeness to planarity for bar k-visibility graphs. We obtain an upper bound on the number of edges in a bar k-visibility graph. As a consequence, we obtain an upper bound of 12 on the chromatic number of bar 1-visibility graphs, and a tight upper bound of 8 on the size of the largest complete bar 1-visibility graph. We also consider the thickness of bar k-visibility graphs, obtaining an upper bound of 4 when k = 1, and a bound that is quadratic in k for k > 1.

Rectangle-visibility representations of bipartite graphs

Abstract The paper considers representations of bipartite graphs as rectangle-visibility graphs , i.e., graphs whose vertices are rectangles in the plane, with adjacency determined by horizontal and vertical visibility. It is shown that, for p ⩽ q , K p , q has a representation with no rectangles having collinear sides if and only if p ⩽ 2 or p = 3 and q ⩽ 4. More generally, it is shown that K p , q is a rectangle-visibility graph if and only if p ⩽ 4. Finally, it is shown that every bipartite rectangle-visibility graph on n ⩾ 4 vertices has at most 4 n − 12 edges.

On Rectangle Visibility Graphs

We study the problem of drawing a graph in the plane so that the vertices of the graph are rectangles that are aligned with the axes, and the edges of the graph are horizontal or vertical lines-of-sight. Such a drawing is useful, for example, when the vertices of the graph contain information that we wish displayed on the drawing; it is natural to write this information inside the rectangle corresponding to the vertex. We call a graph that can be drawn in this fashion a rectangle-visibility graph, or RVG. Our goal is to find classes of graphs that are RVGs. We obtain several results: 1. For 1 ≤ k ≤ 4, k-trees are RVGs. 2. Any graph that can be decomposed into two caterpillar forests is an RVG. 3. Any graph whose vertices of degree four or more form a distance-two independent set is an RVG. 4. Any graph with maximum degree four is an RVG. Our proofs are constructive and yield linear-time layout algorithms.

On the thickness and arboricity of a graph

We prove that the thickness and the arboricity of a graph with e edges are at most ⌊e3 + 32⌋ and ⌈e2⌉, respectively, and that the latter bound is best possible.

Rectangle-visibility Layouts of Unions and Products of Trees

The paper considers representations of unions and products of trees as rectangle-visibility graphs (abbreviated RVGs), i.e., graphs whose vertices are rectangles in the plane, with adjacency determined by horizontal and vertical visibility. Our main results are that the union of any tree (or forest) with a depth-1 tree is an RVG, and that the union of two depth-2 trees and the union of a depth-3 tree with a matching are subgraphs of RVGs. We also show that the cartesian product of two forests is an RVG.

Bar k -visibility graphs: bounds on the number of edges, chromatic number, and thickness: bounds on the number of edges, chromatic number, and thickness

Let S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a one-to-one correspondence between vertices of G and bars in S, such that there is an edge between two vertices in G if and only if there exists an unobstructed vertical line of sight between their corresponding bars. If bars are allowed to see through each other, the graphs representable in this way are precisely the interval graphs. We consider representations in which bars are allowed to see through at most k other bars. Since all bar visibility graphs are planar, we seek measurements of closeness to planarity for bar k-visibility graphs. We obtain an upper bound on the number of edges in a bar k-visibility graph. As a consequence, we obtain an upper bound of 12 on the chromatic number of bar 1-visibility graphs, and a tight upper bound of 8 on the size of the largest complete bar 1-visibility graph. We conjecture that bar 1-visibility graphs have thickness at most 2.

Unit bar-visibility layouts of triangulated polygons

A triangulated polygon is a 2-connected maximal outerplanar graph. A unit bar-visibility graph (UBVG for short) is a graph whose vertices can be represented by disjoint, horizontal, unit-length bars in the plane so that two vertices are adjacent if and only if there is a non-degenerate, unobstructed, vertical band of visibility between the corresponding bars. We give combinatorial and geometric characterizations of the triangulated polygons that are UBVGs. To each triangulated polygon G we assign a character string with the property that G is a UBVG if and only if the string satisfies a certain regular expression. Given a string that satisfies this condition, we describe a linear-time algorithm that uses it to produce a UBV layout of G.

Rectangle-Visibility Representations of Bipartite Graphs

The paper considers representations of bipartite graphs as rectanglevisibility graphs, i.e., graphs whose vertices are rectangles in the plane, with adjacency determined by horizontal and vertical visibility. It is shown that, for p≤q, Kp, q has a representation with no rectangles having collinear sides if and only if p≤3 or p=3 and q≤4. More generally, it is shown that Kp, q is a rectangle-visibility graph if and only if p≤4. Finally, it is shown that every bipartite rectangle-visibility graph on n≥4 vertices has at most 4n−12 edges.

Posets of geometric graphs

A geometric graph Ḡ is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call Ḡ a geometric realization of the underlying abstract graph G . A geometric homomorphism f : Ḡ → H is a vertex map that preserves adjacencies and crossings (but not necessarily non-adjacencies or non-crossings). This work uses geometric homomorphisms to introduce a partial order on the set of isomorphism classes of geometric realizations of an abstract graph G . Set Ḡ ≼ Ĝ if Ḡ and Ĝ are geometric realizations of G and there is a vertex-injective geometric homomorphism f : Ḡ → Ĝ . This paper develops tools to determine when two geometric realizations are comparable. Further, for 3 ≤ n ≤ 6, this paper provides the isomorphism classes of geometric realizations of P n , C n and K n , as well as the Hasse diagrams of the geometric homomorphism posets (resp., P n , C n , K n ) of these graphs. The paper also provides the following results for general n : each of P n and C n has a unique minimal element and a unique maximal element; if k ≤ n then P k (resp., C k ) is a subposet of P n (resp., C n ); and K n contains a chain of length n − 2.

Review of visibility algorithms in the plane by Subir Kumar Ghosh (Cambridge University Press, 2007)

Computational Geometry is a young field, tracing its beginnings to the PhD thesis of Michael Shamos [6] in 1978. Its many applications include, to name a few, computer graphics and imaging, robotics and computer vision, and geometric information systems. Topics studied within computational geometry include arrangements, convex hulls, partitions, triangulation of polygons, Voronoi diagrams, and visibility. One of the most famous theorems in computational geometry is the Art Gallery Theorem, and this theorem also serves as an example of the focus of the book under review. Posed by Victor Klee in 1973, it asks how many stationary guards are required to see all points in an art gallery represented by a simple, n-sided polygon. The answer, given first by Chvatal [3] and later by Fisk [5], using an elegant graph-theoretic proof, is that bn/3c guards are always sufficient and may be necessary. Questions such as this one, of visibility within a polygon, are the subject of Visibility Algorithms in the Plane, by S. Ghosh.