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### Simultaneous Embedding of a Planar Graph and Its Dual on the Grid [article]

C. Erten, S. G. Kobourov
2002 arXiv   pre-print
We provide a linear-time algorithm that simultaneously embeds a 3-connected planar graph and its dual on a (2n-2) by (2n-2) integer grid, where n is the total number of vertices in the graph and its dual  ...  We consider the problem of simultaneously embedding a planar graph and its dual into a small integer grid such that the edges are drawn as straight-line segments and the only crossings are between primal-dual  ...  Introduction In this paper we address the problem of simultaneously drawing a planar graph and its dual on a small integer grid.  ...

### Simultaneous Embedding of a Planar Graph and Its Dual on the Grid [chapter]

Cesim Erten, Stephen G. Kobourov
2002 Lecture Notes in Computer Science
We consider the problem of simultaneously embedding a planar graph and its dual on a small integer grid such that the edges are drawn as straight-line segments and the only crossings are between primal-dual  ...  We provide an O(n) time algorithm that simultaneously embeds a 3-connected planar graph and its dual on a (2n − 2) × (2n − 2) integer grid, where n is the total number of vertices in the graph and its  ...  Introduction In this paper we address the problem of simultaneously drawing a planar graph and its dual on a small integer grid.  ...

### Simultaneous Embedding of a Planar Graph and Its Dual on the Grid

Cesim Erten, Stephen G. Kobourov
2005 Theory of Computing Systems
We consider the problem of simultaneously embedding a planar graph and its dual on a small integer grid such that the edges are drawn as straight-line segments and the only crossings are between primal-dual  ...  We provide an O(n) time algorithm that simultaneously embeds a 3-connected planar graph and its dual on a (2n − 2) × (2n − 2) integer grid, where n is the total number of vertices in the graph and its  ...  Introduction In this paper we address the problem of simultaneously drawing a planar graph and its dual on a small integer grid.  ...

### On Simultaneous Graph Embedding [article]

C. A. Duncan, A. Efrat, C. Erten, S. Kobourov, J.S.B. Mitchell
2002 arXiv   pre-print
We consider the problem of simultaneous embedding of planar graphs.  ...  In particular, we show that without mapping, any number of outerplanar graphs can be embedded simultaneously on an O(n)× O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously  ...  Acknowledgments We would like to thank Anna Lubiw for introducing us to the problem of simultaneous graph embedding.  ...

### On simultaneous planar graph embeddings

Peter Brass, Eowyn Cenek, Cristian A. Duncan, Alon Efrat, Cesim Erten, Dan P. Ismailescu, Stephen G. Kobourov, Anna Lubiw, Joseph S.B. Mitchell
2007 Computational geometry
We consider the problem of simultaneous embedding of planar graphs.  ...  If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n) × O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously  ...  Acknowledgments We would like to thank Ed Scheinerman for stimulating discussions about different variations of the problem and Esther M. Arkin for her proof of Theorem 3 (independent of our work).  ...

### On Simultaneous Planar Graph Embeddings [chapter]

P. Brass, E. Cenek, C. A. Duncan, A. Efrat, C. Erten, D. Ismailescu, S. G. Kobourov, A. Lubiw, J. S. B. Mitchell
2003 Lecture Notes in Computer Science
We consider the problem of simultaneous embedding of planar graphs.  ...  If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n) × O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously  ...  Erten and Kobourov [13] present an algorithm for simultaneously embedding a 3-connected planar graph and its dual on an O(n) × O(n) grid.  ...

### Simultaneous Embedding of Planar Graphs [article]

Thomas Bläsius and Stephen G. Kobourov and Ignaz Rutter
2015 arXiv   pre-print
Recently there has been a great deal of work investigating simultaneous embedding problems both from a theoretical and a practical point of view. We survey recent work on this topic.  ...  Simultaneous embedding is concerned with simultaneously representing a series of graphs sharing some or all vertices.  ...  A Plane Graph and its Dual In a simultaneous drawing of a planar graph and its dual each vertex in the dual graph is required to be placed inside the corresponding face of the primal graph.  ...

### Combining Problems on RAC Drawings and Simultaneous Graph Drawings [chapter]

Evmorfia N. Argyriou, Michael A. Bekos, Michael Kaufmann, Antonios Symvonis
2012 Lecture Notes in Computer Science
This is summarized in the following theorem. Theorem 4. Given a planar embedded graph G, a GDual-GRacSim drawing of G and its dual G * does not always exist.  ...  A closely related problem is the following: Given a planar embedded graph G, determine a geometric drawing of G and its dual G * (without the face-vertex corresponding to the external face) such that:  ...

### Simultaneous Embeddability of Two Partitions [article]

Jan Christoph Athenstädt, Tanja Hartmann, Martin Nöllenburg
2014 arXiv   pre-print
We show that (i) every pair of partitions has a weak simultaneous embedding, (ii) it is NP-complete to decide the existence of a strong simultaneous embedding, and (iii) the existence of a full simultaneous  ...  We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks.  ...  We call G Γ the contour graph of Γ and its dual graph G * Γ the dual graph of Γ . The faces of G Γ belong to zero, one, or two block regions.  ...

### Simultaneous Embeddability of Two Partitions [chapter]

Jan Christoph Athenstädt, Tanja Hartmann, Martin Nöllenburg
2014 Lecture Notes in Computer Science
We show that (i) every pair of partitions has a weak simultaneous embedding, (ii) it is NP-complete to decide the existence of a strong simultaneous embedding, and (iii) the existence of a full simultaneous  ...  We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks.  ...  We thank the anonymous reviewers for helpful comments. References  ...

### Geometric RAC Simultaneous Drawings of Graphs

Evmorfia N. Argyriou, Michael A. Bekos, Michael Kaufmann, Antonios Symvonis
2013 Journal of Graph Algorithms and Applications
goal is to determine a geometric drawing of G and its weak dual G * such that: (i) G and G * are drawn planar, (ii) each vertex of the dual is drawn inside its corresponding face of G and, (iii) the primal-dual  ...  We further show that a cycle and a matching always admit a geometric RAC simultaneous drawing.We also study a closely related problem according to which we are given a planar embedded graph G and the main  ...  The work of M.A.  ...

### Geometric RAC Simultaneous Drawings of Graphs [chapter]

Evmorfia Argyriou, Michael Bekos, Michael Kaufmann, Antonios Symvonis
2012 Lecture Notes in Computer Science
goal is to determine a geometric drawing of G and its weak dual G * such that: (i) G and G * are drawn planar, (ii) each vertex of the dual is drawn inside its corresponding face of G and, (iii) the primal-dual  ...  We further show that a cycle and a matching always admit a geometric RAC simultaneous drawing.We also study a closely related problem according to which we are given a planar embedded graph G and the main  ...  The work of M.A.  ...

### Monotone Simultaneous Embeddings of Upward Planar Digraphs

Oswin Aichholzer, Thomas Hackl, Sarah Lutteropp, Tamara Mchedlidze, Alexander Pilz, Birgit Vogtenhuber
2015 Journal of Graph Algorithms and Applications
We study monotone simultaneous embeddings of upward planar digraphs, which are simultaneous embeddings where the drawing of each digraph is upward planar, and the directions of the upwardness of different  ...  On the other hand, we show that already for three paths, any monotone simultaneous embedding might need a grid whose size is exponential in the number of vertices.  ...  Acknowledgements We thank anonymous referees for helpful comments and for making us aware of the related work [3] .  ...

### Monotone Simultaneous Embedding of Directed Paths [article]

Oswin Aichholzer and Thomas Hackl and Sarah Lutteropp and Tamara Mchedlidze and Alexander Pilz and Birgit Vogtenhuber
2014 arXiv   pre-print
We study monotone simultaneous embeddings of upward planar digraphs, which are simultaneous embeddings where the drawing of each digraph is upward planar, and the directions of the upwardness of different  ...  On the other hand, we show that already for three paths, any monotone simultaneous embedding might need a grid of exponential (w.r.t. the number of vertices) size.  ...  Research on this topic was initiated during a research visit of Sarah Lutteropp and Tamara Mchedlidze in February 2013 in Graz, Austria.  ...

### Level-Planar Drawings with Few Slopes [article]

Guido Brückner, Nadine Davina Krisam, Tamara Mchedlidze
2019 arXiv   pre-print
Moreover, we consider the partial drawing extension problem, where we seek to extend an immutable drawing of a subgraph to a drawing of the whole graph, and the simultaneous drawing problem, which asks  ...  We introduce and study level-planar straight-line drawings with a fixed number λ of slopes.  ...  It is evident that any level-planar two-slope grid drawing of the resulting graph induces a solution of the underlying planar monotone 3-Sat problem and vice versa.  ...
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