Drawing Planar Graphs with Prescribed Face Areas

Kleist, Linda

doi:10.1007/978-3-662-53536-3_14

Cited by 4 publications

(17 citation statements)

References 13 publications

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“…Performing a diamond addition of order on some plane graph changes the degree of exactly two vertices by while all other vertex degrees remain the same. Consequently, if is even, all vertices of K have even degree, and hence, K as an Eulerian triangulation is not area-universal as shown by the author in [15,Theorem 1]. It remains to prove the area-universality of odd accordion graphs with the help of Theorem 1.…”

Section: Accordion Graphsmentioning

confidence: 96%

“…The double stacking graph H ,k is the plane graph obtained from G by applying a diamond addition of order − 1 on Au and a diamond addition of order k − 1 on vw. If · k is odd, every plane graph in [H ,k ] is Eulerian and hence not areauniversal by [15,Theorem 1]. If · k is even, we consider an algebraically independent area assignment of H ,k , show that its last face function is crr-free and has odd max-degree.…”

Section: Double Stacking Graphsmentioning

confidence: 99%

“…Kleist [15] generalized this result by introducing a simple counting argument which shows that no Eulerian triangulation, different from K 3 , is area-universal. Moreover, it is shown in [15] that every 1-subdivision of a plane graphs is areauniversal; that is, every area assignment of a plane graph has a realizing polyline drawing where each edge has at most one bend. Evans et al [10,16] present classes of area-universal plane quadrangulations.…”

Section: Introductionmentioning

confidence: 99%

“…Surprisingly, the insertion of an even number of vertices yields a non-area-universal graph while the insertion of an odd number of vertices yields an area-universal graph. Accordions with an even number of vertices are Eulerian and thus not area-universal [15]. Consequently, area-universal and non-area-universal graphs may have a very similar structure.…”

Section: Introductionmentioning

confidence: 99%

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On the Area-Universality of Triangulations

Kleist

2018

Lecture Notes in Computer Science

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We study straight-line drawings of planar graphs with prescribed face areas. A plane graph is area-universal if for every area assignment on the inner faces, there exists a straight-line drawing realizing the prescribed areas. For triangulations with a special vertex order, we present a sufficient criterion for area-universality that only requires the investigation of one area assignment. Moreover, if the sufficient criterion applies to one plane triangulation, then all embeddings of the underlying planar graph are also area-universal. To date, it is open whether area-universality is a property of a plane or planar graph. We use the developed machinery to present area-universal families of triangulations. Among them we characterize area-universality of accordion graphs showing that area-universal and non-area-universal graphs may be structural very similar. Proposition 2. A plane triangulation T is area-universal if and only if every 4-connected component of T is area-universal.Proof (Sketch). The proof is based on the fact that a plane graph G with a separating triangle t is area-universal if and only if G e , the induced graph by t and its exterior, and G i , the induced graph by t and its interior, are areauniversal. In particular, Lemma 1 allows us to combine realizing drawings of G e and G i to a drawing of G.Remark. Note that a plane 3-tree has no 4-connected component. (Recall that K 4 is 3-connected and a graph on n > 4 vertices is 4-connected if and only if it has no separating triangle.) This is another way to see their area-universality.

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Section: Accordion Graphsmentioning

confidence: 96%

Section: Double Stacking Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Area-Universality of Triangulations

Kleist

2018

Lecture Notes in Computer Science

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“…The area-universal convex polygonal drawings for biconnected outer planar graphs are given in [12]. Recently, it has been shown that a planar graph G is area-universal if for any area assignment to the inner regions of G, a straight line drawing of G can be realized [13,14] and the area-universality of subgraphs of such graphs can be seen in [15]. The extension of the concept of area-universality to rectilinear duals has been given in [16,17].…”

Section: Related Workmentioning

confidence: 99%

Rectangularly Dualizable Graphs: Area-Universality

Kumar¹,

Shekhawat²

2021

Preprint

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A plane graph is called a rectangular graph if each of its edges can be oriented either horizontally or vertically, each of its interior regions is a four-sided region and all interior regions can be fitted in a rectangular enclosure. If the dual of a plane graph is a rectangular graph, then the plane graph is a rectangularly dualizable graph. A rectangular dual is area-universal if any assignment of areas to each of its regions can be realized by a combinatorially weak equivalent rectangular dual. It is still unknown that there exists no polynomial time algorithm to construct an area-universal rectangular dual for a rectangularly dualizable graph . In this paper, we describe a class of rectangularly dualizable graphs wherein each graph can be realized by an areauniversal rectangular dual. We also present a polynomial time algorithm for its construction.

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Completeness for the Complexity Class $$\forall \exists \mathbb {R}$$ and Area-Universality

Dobbins

Kleist

Miltzow

et al. 2022

Discrete Comput Geom

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Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class $$\exists \mathbb {R}$$ ∃ R plays a crucial role in the study of geometric problems. Sometimes $$\exists \mathbb {R}$$ ∃ R is referred to as the ‘real analog’ of NP. While NP is a class of computational problems that deals with existentially quantified boolean variables, $$\exists \mathbb {R}$$ ∃ R deals with existentially quantified real variables. In analogy to $$\Pi _2^p$$ Π 2 p and $$\Sigma _2^p$$ Σ 2 p in the famous polynomial hierarchy, we study the complexity classes $$\forall \exists \mathbb {R}$$ ∀ ∃ R and $$ \exists \forall \mathbb {R}$$ ∃ ∀ R with real variables. Our main interest is the AreaUniversality problem, where we are given a plane graph G, and ask if for each assignment of areas to the inner faces of G, there exists a straight-line drawing of G realizing the assigned areas. We conjecture that AreaUniversality is $$\forall \exists \mathbb {R}$$ ∀ ∃ R -complete and support this conjecture by proving $$\exists \mathbb {R}$$ ∃ R - and $$\forall \exists \mathbb {R}$$ ∀ ∃ R -completeness of two variants of AreaUniversality. To this end, we introduce tools to prove $$\forall \exists \mathbb {R}$$ ∀ ∃ R -hardness and membership. Finally, we present geometric problems as candidates for $$\forall \exists \mathbb {R}$$ ∀ ∃ R -complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.

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Drawing Planar Graphs with Prescribed Face Areas

Cited by 4 publications

References 13 publications

On the Area-Universality of Triangulations

On the Area-Universality of Triangulations

Rectangularly Dualizable Graphs: Area-Universality

Completeness for the Complexity Class $$\forall \exists \mathbb {R}$$ and Area-Universality

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