On rectangle visibility graphs

We prove that the geometric thickness of graphs whose maximum degree is no more than four is two. All of our algorithms run in O(n) time, where n is the number of vertices in the graph. In our proofs, we present an embedding algorithm for graphs with maximum degree three that uses an n × n grid and a more complex algorithm for embedding a graph with maximum degree four. We also show a variation using orthogonal edges for maximum degree four graphs that also uses an n × n grid. The results have implications in graph theory, graph drawing, and VLSI design.

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“…In particular Bose et al [8] show that any graph with maximum degree four is a rectangle visibility graph.…”

Section: Related Workmentioning

confidence: 99%

The geometric thickness of low degree graphs

Duncan

Eppstein

Kobourov

2004

Proceedings of the Twentieth Annual Symposium on Computational Geometry

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“…In practice, our algorithm may produce drawings which clearly prefer one dimension against the other; this is because we use a visibility representation which is biased to one dimension. Recently, some study has been done on the problem of "2-dimensional visibility representations" [7,13] of planar graphs. In a 2-dimensional visibility representation, each vertex is represented by a box and each edge is represented by a horizontal or vertical segment between the sides of the boxes.…”

Section: A Lower Bound For Bendsmentioning

confidence: 99%

Drawing clustered graphs on an orthogonal grid

Eades¹,

Feng

1997

Graph Drawing

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Abstract. Clustered graphs are graphs with recursive clustering structures over the vertices. For graphical representation, the clustering structure is represented by a simple region that contains the drawing of all the vertices which belong to that cluster. In this paper, we present an algorithm which produces planar drawings of clustered graphs in a convention known as orthogonal grid rectangular cluster drawings. We present an algorithm which produces such drawings with O(n 2) area and with at most 3 bends in each edge. This result is as good as existing results for classical planar graphs. Further, we show that our algorithm is optimal in terms of the number of bends in each edge. IntroductionClustered graphs are graphs with recursive clustering structures over the vertices (see Figure 1). Algorithms for constructing straight-line drawings of clustered graphs are given in [8,11]; note, however, that straight-line drawings of clustered graphs can require exponential area [11]. In this paper, we present an algorithm which produce planar drawings of clustered graphs in a convention called "orthogonM grid rectangular cluster drawings". We apply a technique to order the clusters of the graph recursively, and we use the visibility representation for directed graphs to produce our drawings. The orthogonM grid drawing convention appears in a number of applications, such as VLSI circuit design [20,21,37,38] and diagrammatic interfaces for relational information systems [1,2,24,27,31]. Under the orthogonal grid drawing convention, minimizing the number of bends and minimizing the area are the

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“…The problem of determining the visible pairs of n given vertical segments in the horizontal visibility model was studied by Lodi et al in 1986 [5]. They proposed a parallel algorithm to solve the visibility problem among n vertical segments in a plane, considered by Bose et al in 1997 [6]. They studied the problem of computing rectangle visibility graphs (RVGs) of a set of vertical bars [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

S-visibility problem in VLSI chip design

Eskandari¹,

Yeganeh²

2017

Turk J Elec Eng & Comp Sci

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Abstract:In this paper, a new version of very large scale integration (VLSI) layouts compaction problem is considered.Bar visibility graph (BVG) is a simple geometric model for VLSI chip design and layout problems. In all previous works, vertical bars or other chip components in the plane model gates, as well as edges, are modeled by horizontal visibilities between bars. In this study, for a given set of vertical bars, the edges can be modeled with orthogonal paths known as staircases. Therefore, we consider a new version of bar visibility graphs (BsVG). We then present an algorithm to solve the s-visibility problem of vertical segments, which can be implemented on a VLSI chip. Our algorithm determines all the pairs of segments that are s-visible from each other.

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On rectangle visibility graphs

Cited by 33 publications

References 10 publications

The geometric thickness of low degree graphs

The geometric thickness of low degree graphs

Drawing clustered graphs on an orthogonal grid

S-visibility problem in VLSI chip design

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