The SPQR-Tree Data Structure in Graph Drawing

Mutzel, Petra

doi:10.1007/3-540-45061-0_4

Cited by 11 publications

(9 citation statements)

References 24 publications

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“…The planar embeddings of a 2-connected graph can be represented by the SPQR-tree, which is a data structure introduced by Di Battista and Tamassia [3,4]. A more detailed description of the SPQR-tree can be found in the literature [3,4,5,12]. Here we just give a sketch and some notation.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

A Kuratowski-type theorem for planarity of partially embedded graphs

Jelínek¹,

Kratochvíl²,

Rutter³

2013

Computational Geometry

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A partially embedded graph (or Peg) is a triple (G, H, H), where G is a graph, H is a subgraph of G, and H is a planar embedding of H. We say that a Peg (G, H, H) is planar if the graph G has a planar embedding that extends the embedding H.We introduce a containment relation of Pegs analogous to graph minor containment, and characterize the minimal non-planar Pegs with respect to this relation. We show that all the minimal non-planar Pegs except for finitely many belong to a single easily recognizable and explicitly described infinite family. We also describe a more complicated containment relation which only has a finite number of minimal non-planar Pegs.Furthermore, by extending an existing planarity test for Pegs, we obtain a polynomialtime algorithm which, for a given Peg, either produces a planar embedding or identifies an obstruction.

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Section: Preliminaries and Notationmentioning

confidence: 99%

A Kuratowski-type theorem for planarity of partially embedded graphs

Jelínek¹,

Kratochvíl²,

Rutter³

2013

Computational Geometry

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“…This notion was introduced in [4] to describe all the possible embeddings of 2-connected planar graphs in a succinct way and was used in various situations when asking for planar embeddings with special properties. A survey on the use of this technique in planar graphs is [22]. It is indeed obvious that if a 2-connected graph admits a feasible drawing, then the skeleton of each node of the SPQR-tree has a drawing compatible (a precise definition of compatibility will come later) with the partial embedding.…”

Section: Introductionmentioning

confidence: 99%

Testing Planarity of Partially Embedded Graphs

Angelini¹,

Battista²,

Frati³

et al. 2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

112

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We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the "oncas" behaviour -obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently.Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach linear-time complexity.Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NP-hard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.

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“…A more detailed description of the SPQR-tree can be found in the literature [3,4,5,11]. Here we just give a sketch and some notation.…”

Section: Connectivity and Spqr-treesmentioning

confidence: 99%

A kuratowski-type theorem for planarity of partially embedded graphs

Jelínek

Kratochvíl

Rutter

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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A partially embedded graph (or Peg) is a triple (G, H, H), where G is a graph, H is a subgraph of G, and H is a planar embedding of H. We say that a Peg (G, H, H) is planar if the graph G has a planar embedding that extends the embedding H.We introduce a containment relation of Pegs analogous to graph minor containment, and characterize the minimal non-planar Pegs with respect to this relation. We show that all the minimal non-planar Pegs except for finitely many belong to a single easily recognizable and explicitly described infinite family. We also describe a more complicated containment relation which only has a finite number of minimal non-planar Pegs.Furthermore, by extending an existing planarity test for Pegs, we obtain a polynomial-time algorithm which, for a given Peg, either produces a planar embedding or identifies a minimal obstruction.

show abstract

The SPQR-Tree Data Structure in Graph Drawing

Cited by 11 publications

References 24 publications

A Kuratowski-type theorem for planarity of partially embedded graphs

A Kuratowski-type theorem for planarity of partially embedded graphs

Testing Planarity of Partially Embedded Graphs

A kuratowski-type theorem for planarity of partially embedded graphs

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