## Radial Level Planarity Testing and Embedding in Linear Time (2005)

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Venue: | Journal of Graph Algorithms and Applications |

Citations: | 20 - 9 self |

### BibTeX

@ARTICLE{Bachmaier05radiallevel,

author = {Christian Bachmaier and Franz J. Brandenburg and Michael Forster},

title = {Radial Level Planarity Testing and Embedding in Linear Time},

journal = {Journal of Graph Algorithms and Applications},

year = {2005},

volume = {9},

pages = {2005}

}

### Years of Citing Articles

### OpenURL

### Abstract

A graph with a given partition of the vertices on k concentric circles is radial level planar if there is a vertex permutation such that the edges can be routed strictly outwards without crossings. Radial level planarity extends level planarity, where the vertices are placed on k horizontal lines and the edges are routed strictly downwards without crossings. The extension is characterised by rings, which are level non-planar biconnected components. Our main results are linear time algorithms for radial level planarity testing and for computing an embedding. We introduce PQR-trees as a new data structure where R-nodes and associated templates for their manipulation are introduced to deal with rings. Our algorithms extend level planarity testing and embedding algorithms which use PQ-trees.

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Citation Context ...ph after a positive level planarity test and for a level planar drawing the algorithm computes a level embedding in two passes. It is outlined in Algorithm 6. First G is augmented to a planar stgraph =-=[17,29]-=-. This is a biconnected graph with two adjacent vertices s and t and a bijective numbering st : V → {1,...,|V |} of the vertices such that st(s) = 1, st(t) = |V |, and that for every vertex v with 1 <... |

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Citation Context ...partition. Such a partition may be due to application-specific attributes of the graph (e. g. hierarchies as in [23]) or it may be introduced by structural evaluation (e. g. centrality measures as in =-=[6, 7]-=-) or by the drawing algorithm (e. g. the Sugiyama framework [12,33]). Formally, we are given a k-level graph G = (V,E,φ) with a level assignment φ: V → {1,2,...,k} with k ≤ |V | that partitions the ve... |

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Citation Context ...e. As is the case for arbitrary drawings empirical evidence suggests that the number of crossings is a major factor for readability of levelled drawings [31]. The (horizontal) level planarity problem =-=[13,20,26]-=- is the question whether or not a level graph G can be drawn in the plane such that all vertices of the j-th level V j are placed on the j-th horizontal line lj = {(x,j) | x ∈ R } and the edges are dr... |

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Citation Context ...are based on the level planarity testing algorithm of Jünger, Leipert, and Mutzel [24–26, 28], in the following called the JLM algorithm, which in turn is based on the approach of Heath and Pemmaraju =-=[20,21]-=-. All these algorithms extend the level planarity testing algorithm of Di Battista and Nardelli [13] to arbitrary level graphs. Previously only hierarchies could be treated, which are level graphs wit... |

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Citation Context ...adial level planar graphs. They are based on PQ-trees and contain a new “R” node type for the rings. PQR-trees are not related to SPQR-trees that are used for example in incremental planarity testing =-=[14]-=-. 4.1 R-Nodes R-nodes are similar to Q-nodes. Their new properties express the differences between rings and other biconnected components. An R-node is drawn as an elliptical ring. The admissible oper... |

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Citation Context ...e. As is the case for arbitrary drawings empirical evidence suggests that the number of crossings is a major factor for readability of levelled drawings [31]. The (horizontal) level planarity problem =-=[13,20,26]-=- is the question whether or not a level graph G can be drawn in the plane such that all vertices of the j-th level V j are placed on the j-th horizontal line lj = {(x,j) | x ∈ R } and the edges are dr... |

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Citation Context ...to decide whether a graph is radial level planar in Section 4. The computation of an embedding is described in Section 5. We conclude with a summary. 2 Level Planarity 2.1 Foundations Dujmović et al. =-=[15]-=- have applied the concept of fixed parameter tractability to graph drawing. It can be shown that k-level planarity and radial k-level planarity are fixed parameter tractable. However, k must be bounde... |

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Citation Context ...ations B–D. The contacts store which sinks have to be augmented if the new introduced Q-node is inserted into its parent Q-node by an application of a template later in the algorithm. For details see =-=[24,25,28]-=-.C. Bachmaier et al., Radial Level Planarity, JGAA, 9(1) 53–97 (2005) 67 3 3 2 1 2 1 8 2 5 1 2 1 7 6 4 4 (a) (b) Figure 4: A radial level planar connected component and a radial level nonplanar graph... |

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Citation Context ...e. As is the case for arbitrary drawings empirical evidence suggests that the number of crossings is a major factor for readability of levelled drawings [31]. The (horizontal) level planarity problem =-=[13,20,26]-=- is the question whether or not a level graph G can be drawn in the plane such that all vertices of the j-th level V j are placed on the j-th horizontal line lj = {(x,j) | x ∈ R } and the edges are dr... |

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Citation Context ...he linear time algorithm of Chandramouli and Diwan [8] determines whether a triconnected DAG is level planar. Because the JLM algorithm is rather involved and difficult to implement, Healy and Kuusik =-=[18]-=- have presented a simpler approach for the detection of level planarity. Their algorithm runs in O(|V | 2 ) time for proper level graphs and O(|V | 4 ) time in the general case. If an embedding is nee... |

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Citation Context ... k-level planar if there are permutations of the vertices on each radial level such that the edges can be drawn as strictly monotone curves from inner to outer levels without crossings. Such drawings =-=[3]-=- extend the radial tree drawings of Eades [16], where the levels of the vertices are given by their depth, i. e., BFS-level. Figure 1(b) shows a radial level planar drawing of the graph in Figure 1(a)... |

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Citation Context ...ource. A source is a vertex with edges only to vertices on higher levels, whereas a sink is a vertex with edges only from vertices on lower levels. The linear time algorithm of Chandramouli and Diwan =-=[8]-=- determines whether a triconnected DAG is level planar. Because the JLM algorithm is rather involved and difficult to implement, Healy and Kuusik [18] have presented a simpler approach for the detecti... |

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Citation Context ...ations B–D. The contacts store which sinks have to be augmented if the new introduced Q-node is inserted into its parent Q-node by an application of a template later in the algorithm. For details see =-=[24,25,28]-=-.C. Bachmaier et al., Radial Level Planarity, JGAA, 9(1) 53–97 (2005) 67 3 3 2 1 2 1 8 2 5 1 2 1 7 6 4 4 (a) (b) Figure 4: A radial level planar connected component and a radial level nonplanar graph... |

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Citation Context ...ion We have presented new linear time algorithms for testing radial level planarity and computing radial level planar embeddings. They can easily be extended to circle planarity testing and embedding =-=[2]-=-, where edges having both vertices on the same level are allowed. Further investigations are required to expand the test algorithms for level planarity for detecting the so called minimum level non-pl... |

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Citation Context ...for level planarity for detecting the so called minimum level non-planar subgraph patterns (MLNP-patterns) if the tested graph is level non-planar. MLNP-patterns for level graphs are characterised in =-=[19]-=- and are the counterparts of the Kuratowski Graphs K3,3 and K5 for graph planarity which can efficiently be computed [27, 30, 35]. Similar patterns for the radial case are desirable, see [1] for first... |

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Citation Context ...| 2 ) time for proper level graphs and O(|V | 4 ) time in the general case. If an embedding is needed, the time complexity raises to O(|V | 3 ) and O(|V | 6 ), respectively. Finally, Randerath et al. =-=[32]-=- have presented a quadratic time reduction of level planarity of proper level graphs to the satisfiability problem of Boolean formulas in 2CNF, which is solvable in linear time. Our algorithm is based... |

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Citation Context ...ime for an arbitrary number of levels. Our algorithms are practical and have been realised in a prototypical implementation in C++ using the Graph Template Library (GTL) with improved symmetric lists =-=[4]-=-. They are based on the level planarity testing algorithm of Jünger, Leipert, and Mutzel [24–26, 28], in the following called the JLM algorithm, which in turn is based on the approach of Heath and Pem... |

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Citation Context ...ph after a positive level planarity test and for a level planar drawing the algorithm computes a level embedding in two passes. It is outlined in Algorithm 6. First G is augmented to a planar stgraph =-=[17,29]-=-. This is a biconnected graph with two adjacent vertices s and t and a bijective numbering st : V → {1,...,|V |} of the vertices such that st(s) = 1, st(t) = |V |, and that for every vertex v with 1 <... |

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Citation Context ...graph is level non-planar. MLNP-patterns for level graphs are characterised in [19] and are the counterparts of the Kuratowski Graphs K3,3 and K5 for graph planarity which can efficiently be computed =-=[27, 30, 35]-=-. Similar patterns for the radial case are desirable, see [1] for first steps in that direction. As already mentioned in the conclusion of [28, p. 211] the detection of MLNP-patterns can also be used ... |

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Citation Context ...s [12]. The number of levels spanned by long edges may be linear in the size of the graph, as a nested sequence of triangles shows [10,11]. However, every planar graph has a concentric representation =-=[34]-=- based on a breadth first search (BFS) traversal, if in addition to the levelling the monotonicity of the edges is discarded. There the vertices are placed on concentric circles corresponding to BFS-l... |

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Citation Context ...rised in [19] and are the counterparts of the Kuratowski Graphs K3,3 and K5 for graph planarity which can efficiently be computed [27, 30, 35]. Similar patterns for the radial case are desirable, see =-=[1]-=- for first steps in that direction. As already mentioned in the conclusion of [28, p. 211] the detection of MLNP-patterns can also be used to verify the results of a (radial) level planarity test. Sin... |

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Citation Context ...4 1 Introduction We consider the problem of drawing a graph with a given ordered vertex partition. Such a partition may be due to application-specific attributes of the graph (e. g. hierarchies as in =-=[23]-=-) or it may be introduced by structural evaluation (e. g. centrality measures as in [6, 7]) or by the drawing algorithm (e. g. the Sugiyama framework [12,33]). Formally, we are given a k-level graph G... |