Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems

Kant, Goos; He, Xin

doi:10.1016/s0304-3975(95)00257-x

Cited by 150 publications

(147 citation statements)

References 14 publications

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“…Transversal structures have been studied in [KH97], [Fus07], and [Fus09]. The relevance of transversal structures in our context comes from the following simple proposition.…”

Section: Square Duals and Transversal Structuresmentioning

confidence: 98%

“…An elegant approach to proving the theorem is to split the task into two: In the first step it is shown that the graph can be enriched with some combinatorial structure, in the second step this structure is used to construct the geometric representation. Such an approach was taken by Bhasker and Sahni [BS88] and later refined by He [He93], Kant and He [KH97] and Fusy [Fus07]. The later of these papers use transversal structures (c.f., Subsection 5.1) as the intermediate combinatorial structure.…”

Section: Rectangular Dualmentioning

confidence: 99%

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Rectangle and Square Representations of Planar Graphs

Felsner

2012

Thirty Essays on Geometric Graph Theory

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In the first part of this survey we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the rectangular dual. In visibility graphs and segment contact graphs the vertices correspond to horizontal or to horizontal and vertical segments of the dissection. Special orientations of graphs turn out to be helpful when dealing with characterization and representation questions. Therefore, we look at orientations with prescribed degrees, bipolar orientations, separating decompositions, and transversal structures.In the second part we ask for representations by a dissections of a rectangle into squares. We review results by Brooks et al. (1940), Kenyon (1998) and Schramm (1993) and discuss a technique of computing squarings via solutions of systems of linear equations.Mathematics Subject Classifications (2010) 05C10, 05C62, 52C15.

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“…Transversal structures have been studied in [KH97], [Fus07], and [Fus09]. The relevance of transversal structures in our context comes from the following simple proposition.…”

Section: Square Duals and Transversal Structuresmentioning

confidence: 98%

Section: Rectangular Dualmentioning

confidence: 99%

Rectangle and Square Representations of Planar Graphs

Felsner

2012

Thirty Essays on Geometric Graph Theory

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“…A common way to measure this is by the smallest integer grid in which a drawing can be realized (see, e.g. [4,[32][33][34]48] for papers that specifically study the area required by visibility representations of graphs and Di Battista and Frati [13] for a survey on compact drawings of graphs).…”

Section: Theoremmentioning

confidence: 99%

The Partial Visibility Representation Extension Problem

et al. 2017

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(u, v) of G. We study a generalization of the recognition problem where a function ψ defined on a subset V of V (G) is given and the question is whether there is a bar visibility representation ψ of G with ψ(v) = ψ (v) for every v ∈ V . We show that for undirected graphs this problem, and other closely related problems, is NP-complete, but for certain cases involving directed graphs it is solvable in polynomial time.

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“…Here, we are given a graph G = (V, E) on n vertices and m edges and an integer b and asked whether the imbalance of G is at most b. The problem finds a variety of applications in graph drawing [9,10,16,19,20]. The computational aspects of graph imbalance were studied by Biedl et al [2], who show that computing the imbalance of G is NP-hard.…”

Section: Introductionmentioning

confidence: 99%

Imbalance Is Fixed Parameter Tractable

Lokshtanov

Misra

Saurabh

2010

Lecture Notes in Computer Science

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Abstract. In the Imbalance Minimization problem we are given a graph G = (V, E) and an integer b and asked whether there is an ordering v1 . . . vn of V such that the sum of the imbalance of all the vertices is at most b. The imbalance of a vertex vi is the absolute value of the difference between the number of neighbors to the left and right of vi. The problem is also known as the Balanced Vertex Ordering problem and it finds many applications in graph drawing. We show that this problem is fixed parameter tractable and provide an algorithm that runs in time 2 O(b log b) · n O(1) . This resolves an open problem of Kára et al. [COCOON 2005].

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Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems

Cited by 150 publications

References 14 publications

Rectangle and Square Representations of Planar Graphs

Rectangle and Square Representations of Planar Graphs

The Partial Visibility Representation Extension Problem

Imbalance Is Fixed Parameter Tractable

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