Characterization of unlabeled level planar trees

Estrella-Balderrama, Alejandro; Fowler, J. Joseph; Kobourov, Stephen G.

doi:10.1016/j.comgeo.2008.12.006

Cited by 25 publications

(41 citation statements)

References 10 publications

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“…When regions are labelled lines in the plane, Estrella-Balderrama [EBFK09] showed that for every n ≥ 6, there is no set of n parallel lines in the plane that support all labelled n-vertex planar graphs. The authors moreover characterized a (sub)class of n-vertex planar graphs that are supported by every set of n-parallel lines, for every labelling of the graphs in the class.…”

Section: Introductionmentioning

confidence: 99%

A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Dujmović

Langerman

2012

Discrete Comput Geom

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Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in R 2 , there is a line such that in both line sets, for both halfplanes delimited by , there are √ n lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are n/3 of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erdős-Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to Graph Drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labelled lines that are universal for all n-vertex labelled planar graphs. As a side note, we prove that every set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar graphs.

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Section: Introductionmentioning

confidence: 99%

A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Dujmović

Langerman

2012

Discrete Comput Geom

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“…Proof. The labelings of T 8 and T 9 were shown not to have planar realizations in [5]. We need to do the same for the labelings of the remaining five graphs in F given in Figure 9.…”

Section: Forbidden Graphsmentioning

confidence: 99%

“…Estrella et al [5] presented linear time recognition algorithms for the class of ULP trees. Providing the equivalent algorithms for general ULP graphs remains for future work.…”

Section: Previous and Future Workmentioning

confidence: 99%

“…Showing that the removal of any edge from T 8 or T 9 yielded a caterpillar, radius-2 star, or degree-3 spider, all members of P, was done in [5]. For G 5 in which a--b--d--e--c--a, a--b--c--a, b--c--d--b, c--d--e--c all form cycles shown in Fig. 9(a), the removal of edges b--c or c--d forms a 2-C E 3-S, while removing of any other edge forms a GC.…”

Section: Lemma 13 Each Forbidden Graph Is Minimal In That the Removmentioning

confidence: 99%

“…While Geyer et al [7] have shown this cannot always be done for tree-tree pairs, the question remains open for tree-path pairs. Estrella et al [5] partially answer this question by characterizing the set of trees that have a simultaneous geometric embedding with a strictly monotone path. We now extend those results by characterizing the set of all planar graphs that have a simultaneous geometric embedding with a strictly monotone path.…”

Section: Introductionmentioning

confidence: 99%

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Characterization of Unlabeled Level Planar Graphs

Fowler

Kobourov

Lecture Notes in Computer Science

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Abstract. We present the set of planar graphs that always have a simultaneous geometric embedding with a strictly monotone path on the same set of n vertices, for any of the n! possible mappings. These graphs are equivalent to the set of unlabeled level planar (ULP) graphs that are level planar over all possible labelings. Our contributions are twofold. First, we provide linear time drawing algorithms for ULP graphs. Second, we provide a complete characterization of ULP graphs by showing that any other graph must contain a subgraph homeomorphic to one of seven forbidden graphs.

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Planar Projections of Graphs

Aravind

Maniyar

2020

Algorithms and Discrete Applied Mathematics

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We introduce and study a new graph representation where vertices are embedded in three or more dimensions, and in which the edges are drawn on the projections onto the axis-parallel planes. We show that the complete graph on n vertices has a representation in n/2+1 planes. In 3 dimensions, we show that there exist graphs with 6n − 15 edges that can be projected onto two orthogonal planes, and that this is best possible. Finally, we obtain bounds in terms of parameters such as geometric thickness and linear arboricity. Using such a bound, we show that every graph of maximum degree 5 has a plane-projectable representation in 3 dimensions.

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Characterization of unlabeled level planar trees

Cited by 25 publications

References 10 publications

A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Characterization of Unlabeled Level Planar Graphs

Planar Projections of Graphs

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