Embedding Vertices at Points: Few Bends Suffice for Planar Graphs

Kaufmann, Michael; Wiese, Roland

doi:10.1142/9789812796608_0005

Cited by 12 publications

(23 citation statements)

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“…Let G be a planar 2-colored graph and let S = B ∪ R be a set of points in the plane, such that |B| = |V b | and |R| = |V r |. We call blue points the points of B and red points the points of R. A point-set embedding onto S of G is a planar drawing Γ such that the vertices of G are drawn in Γ on the points of S, and each edge of G is drawn as a polyline in Γ (Kaufmann and Wiese [6] show that any planar graph admits a points-set embedding). G has a bichromatic point-set embedding onto S if G has a point-set embedding onto S such that every blue vertex is drawn on a blue point, and every red vertex is drawn on a red point.…”

Section: Preliminariesmentioning

confidence: 99%

How to Embed a Path onto Two Sets of Points

2006

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Let R and B be two sets of points such that the points of R are colored red and the points of B are colored blue. Let P be a path such that |R| vertices of P are red and |B| vertices of P are blue. We study the problem of computing a crossing-free drawing of P such that each blue vertex is represented as a point of B and each red vertex of P is represented as a point of R. We show that such a drawing can always be realized by using at most one bend per edge.

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Section: Preliminariesmentioning

confidence: 99%

How to Embed a Path onto Two Sets of Points

2006

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“…Then construct any planar drawing Γ 1 of G 1 , and let

P (I)

be the point set on which the vertices of I are mapped in Γ 1 . Finally, construct a planar drawing Γ 2 of any

⌈ n / 4 ⌉

‐vertex planar graph G 2 on point set

P (I)

(e.g., using Kaufmann and Wiese's technique ). Since I is an independent set, any bijective mapping between the vertex set of G 2 and I ensures that G 1 and G 2 share no edges.…”

Section: Introductionmentioning

confidence: 99%

SEFE without Mapping via Large Induced Outerplane Graphs in Plane Graphs

Angelini

Evans

Frati

et al. 2015

Journal of Graph Theory

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We show that every n-vertex planar graph admits a simultaneous embedding without mapping and with fixed edges with any (n/2)-vertex planar graph. In order to achieve this result, we prove that every n-vertex plane graph has an induced outerplane subgraph containing at least n/2 vertices. Also, we show that every n-vertex planar graph and every n-vertex planar partial 3-tree admit a simultaneous embedding without mapping and with fixed edges.

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“…For k-bend drawing without mapping, Kaufmann and Wiese [19] gave a 1-bend drawing algorithm for 4-connected plane graphs and a 2-bend drawing algorithm for general plane graphs; for general graphs, their algorithm takes time O(n log n) (resp. O(n 2 )) to draw a point-set embedding with at most three (resp.…”

Section: Introductionmentioning

confidence: 99%

Kinetic and Stationary Point-Set Embeddability for Plane Graphs

2013

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We investigate a kinetic version of point-set embeddability. Given a plane graph G(V, E) where |V | = n, and a set P of n moving points where the trajectory of each point is an algebraic function of constant maximum degree s, we maintain a point-set embedding of G on P with at most three bends per edge during the motion. This requires reassigning the mapping of vertices to points from time to time. Our kinetic algorithm uses linear size, O(n log n) preprocessing time, and processes O(n 2 β2s+2(n) log n) events, each in O(log 2 n) time. Here, βs(n) = λs(n)/n is an extremely slow-growing function and λs(n) is the maximum length of Davenport-Schinzel sequences of order s on n symbols.

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Embedding Vertices at Points: Few Bends Suffice for Planar Graphs

Cited by 12 publications

References 0 publications

How to Embed a Path onto Two Sets of Points

How to Embed a Path onto Two Sets of Points

SEFE without Mapping via Large Induced Outerplane Graphs in Plane Graphs

Kinetic and Stationary Point-Set Embeddability for Plane Graphs

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