Universal Point Sets for Drawing Planar Graphs  with Circular Arcs

Angelini, Patrizio; Eppstein, David; Frati, Fabrizio; Kaufmann, Michael; Lazard, Sylvain; Mchedlidze, Tamara; Teillaud, Monique; Wolff, Alexander

doi:10.7155/jgaa.00324

Cited by 12 publications

(11 citation statements)

References 25 publications

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“…By Fáry's theorem, every planar graph can be drawn planarly with straight line segments for its edges, and therefore it can also be drawn with circular arcs. Indeed, there exist universal sets of n points such that every n-vertex planar graph can be drawn using those points for vertices and circular arcs for edges, something that is not true for straight-line drawing [2]. However, those arcs may not necessarily meet at equal angles.…”

Section: Lombardi Drawingmentioning

confidence: 99%

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A Möbius-Invariant Power Diagram and Its Applications to Soap Bubbles and Planar Lombardi Drawing

Eppstein

2014

Discrete Comput Geom

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We use three-dimensional hyperbolic geometry to define a form of power diagram for systems of circles in the plane that is invariant under Möbius transformations. By applying this construction to circle packings derived from the KoebeAndreev-Thurston circle packing theorem, we show that every planar graph of maximum degree three has a planar Lombardi drawing (a drawing in which the edges are drawn as circular arcs, meeting at equal angles at each vertex). We use circle packing to construct planar Lombardi drawings of a special class of 4-regular planar graphs, the medial graphs of polyhedral graphs, and we show that not every 4-regular planar graph has a planar Lombardi drawing. We also use these power diagrams to characterize the graphs formed by two-dimensional soap bubble clusters (in equilibrium configurations) as being exactly the 3-regular bridgeless planar multigraphs, and we show that soap bubble clusters in stable equilibria must in addition be 3-connected.

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Section: Lombardi Drawingmentioning

confidence: 99%

“…Let be the line through the three centers, assumed to exist by (2). Then because passes through each circle center, a reflection across is a symmetry of each circle and therefore of the whole configuration of three circles.…”

Section: Fig 17mentioning

confidence: 99%

A Möbius-Invariant Power Diagram and Its Applications to Soap Bubbles and Planar Lombardi Drawing

Eppstein

2014

Discrete Comput Geom

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“…If bends may be placed freely in the plane, then a construction of Everett et al using n points is universal for planar graphs [19]. Angelini et al constructed universal point sets of size n, lying on a parabolic path, for planar graphs where the edges are drawn as circular arcs [2]. If edges may consist of two smoothly connected semicircles, then n collinear points are universal for planar graphs [6,22].…”

Section: Introductionmentioning

confidence: 99%

Small Superpatterns for Dominance Drawing

Bannister¹,

Devanny²,

Eppstein³

2013

2014 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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We exploit the connection between dominance drawings of directed acyclic graphs and permutations, in both directions, to provide improved bounds on the size of universal point sets for certain types of dominance drawing and on superpatterns for certain natural classes of permutations. In particular we show that there exist universal point sets for dominance drawings of the Hasse diagrams of width-two partial orders of size O(n 3/2 ), universal point sets for dominance drawings of st-outerplanar graphs of size O(n log n), and universal point sets for dominance drawings of directed trees of size O(n 2 ). We show that 321-avoiding permutations have superpatterns of size O(n 3/2 ), riffle permutations (321-, 2143-, and 2413-avoiding permutations) have superpatterns of size O(n), and the concatenations of sequences of riffles and their inverses have superpatterns of size O(n log n). Our analysis includes a calculation of the leading constants in these bounds.

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“…Very recently, Löffler and Tóth [24] showed that 6n − 10 points suffice for k = 1. Angelini et al [4] showed that n points suffice when we draw the edges as circular arcs.…”

Section: Theoremmentioning

confidence: 99%

“…See Figure 2. 4 to four points in convex position (on the left), then π(T ) has a crossing for all T ∈ T n . Otherwise, π(T ) is plane for at most one T ∈ T n (on the right).…”

Section: Large Universal Point Setsmentioning

confidence: 99%

On Universal Point Sets for Planar Graphs

Cardinal

Hoffmann

Kusters

2013

Computational Geometry and Graphs

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A set P of points in R 2 is n-universal if every planar graph on n vertices admits a plane straight-line embedding on P . Answering a question by Kobourov, we show that there is no n-universal point set of size n, for any n ≥ 15. Conversely, we use a computer program to show that there exist universal point sets for all n ≤ 10 and to enumerate all corresponding order types. Finally, we describe a collection G of 7 393 planar graphs on 35 vertices that do not admit a simultaneous geometric embedding without mapping, that is, no set of 35 points in the plane supports a plane straight-line embedding of all graphs in G.

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Universal Point Sets for Drawing Planar Graphs with Circular Arcs

Abstract: We prove that there exists a set S of n points in the plane such that every n-vertex planar graph G admits a planar drawing in which every vertex of G is placed on a distinct point of S and every edge of G is drawn as a circular arc.

Cited by 12 publications

References 25 publications

A Möbius-Invariant Power Diagram and Its Applications to Soap Bubbles and Planar Lombardi Drawing

A Möbius-Invariant Power Diagram and Its Applications to Soap Bubbles and Planar Lombardi Drawing

Small Superpatterns for Dominance Drawing

On Universal Point Sets for Planar Graphs

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