## Upward Planar Drawing of Single Source Acyclic Digraphs (1990)

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Citations: | 15 - 0 self |

### BibTeX

@MISC{Hutton90upwardplanar,

author = {Michael D. Hutton},

title = {Upward Planar Drawing of Single Source Acyclic Digraphs},

year = {1990}

}

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### Abstract

A upward plane drawing of a directed acyclic graph is a straight line drawing in the Euclidean plane such that all directed arcs point upwards. Thomassen [30] has given a non-algorithmic, graph-theoretic characterization of those directed graphs with a single source that admit an upward drawing. We present an efficient algorithm to test whether a given single-source acyclic digraph has a plane upward drawing and, if so, to find a representation of one such drawing. The algorithm decomposes the graph into biconnected and triconnected components, and defines conditions for merging the components into an upward drawing of the original graph. For the triconnected components we provide a linear algorithm to test whether a given plane representation admits an upward drawing with the same faces and outer face, which also gives a simpler (and algorithmic) proof of Thomassen's result. The entire testing algorithm (for general single source directed acyclic graphs) operates in O(n²) time and...

### Citations

2541 | The Design and Analysis of Computer Algorithms - Aho, Hopcroft, et al. - 1974 |

1071 | Depth-first search and linear graph algorithms
- Tarjan
- 1972
(Show Context)
Citation Context ...oles of E and F reversed. 6.4. The Algorithm. Here we briefly summarize our algorithm and discuss its complexity. Given DAG G, we first isolate biconnected components using the algorithm discussed in =-=[23]-=-. By Lemma 6.1 these can be tested independently. This step requires \Theta(n) time, plus the time to test each piece independently. For a biconnected DAG G, we use the triconnected components algorit... |

480 |
Testing for the consecutive ones property, interval graphs and graph planarity using PQ-tree algorithms
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- 1976
(Show Context)
Citation Context ...on some number of subgraphs of G. The total time is thus quadratic. Algorithm TRI-EMBED: Compute the faces of the tri-connected input graph G and a planar embedding of G using a linear time algorithm =-=[18, 4]-=-. For each face f adjacent to the source, transform the abstract embedding G CHAPTER 7. AN ALGORITHM FOR UPWARD PLANAR EMBEDDING 58 of G to have outer face f and test for strongly equivalent upward pl... |

236 | Efficient planarity testing - Hopcroft, Tarjan - 1974 |

198 |
Embedding planar graphs on the grid
- Schnyder
- 1990
(Show Context)
Citation Context ... to ask how big the integer grid must be or, equivalently, how much real precision is required. If bends are allowed an O(n) by O(n) grid suffices [7], similar to the case of undirected planar graphs =-=[4, 20]-=-. There is no upper bound known on the worst case size requirement for straight-line upward plane drawings, but Di Battista, Tamassia and Tollis [8] have exhibited a class of upward planar digraphs re... |

185 |
Dividing a graph into triconnected components
- Hopcroft, Tarjan
- 1973
(Show Context)
Citation Context ...all edges between u; v and vertices of H. The edge (u; v), if it exists, forms a component by itself. An algorithm for finding triconnected components 1 in linear time is given in Hopcroft and Tarjan =-=[13]. A relate-=-d concept is that of graph/digraph union: we define G 1 [G 2 , for components with "shared" vertices to be the inclusive union of all vertices and edges. That is, for v in both G 1 and G 2 ,... |

127 | On straight lines representation of planar graphs - Fary - 1948 |

112 |
A linear algorithm for embedding planar graphs using PQ-trees
- Chiba, Nishizeki, et al.
- 1985
(Show Context)
Citation Context ...ing can have exponential grid-size, but the polygonal grid drawing guarantees no more than O(n) bends in the polygonal edges and a linear size grid. The same result is achieved by Chiba and Nishizeki =-=[6, 23]-=- as a linear extension of Booth and Lueker's PQ-tree algorithm. The algorithm assumes an st-numbering which also realizes the partial order implied by the digraph G, and modifies the PQ-tree algorithm... |

100 |
Bemerkungen zum Vierfarbenproblem
- Wagner
- 1936
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Citation Context ...5, 68R05, 68R10 1. Introduction. There are a wide range of results dealing with drawing, representing, or testing planarity of graphs. Steinitz and Rademacher [22], F'ary [10], Stein [21], and Wagner =-=[26]-=- independently showed that every planar graph can be drawn in the plane using only straight line segments for the edges. Tutte [25] showed that every 3-connected planar graph admits a convex straight-... |

85 | On the computational complexity of upward and rectilinear planarity testing
- Garg, Tamassia
(Show Context)
Citation Context ...and, if so, to find a representation of one such drawing. This result is made more significant in light of the recent proof, by Garg and Tamassia, that the problem is NP-complete for general digraphs =-=[12]-=-. The algorithm decomposes the digraph into biconnected and triconnected components, and defines conditions for merging the components into an upward plane drawing of the original digraph. To handle t... |

80 | Convex Representations of Graphs - Tutte - 1960 |

60 |
Algorithms for plane representations of acyclic digraphs
- DiBattista, Tamassia
- 1988
(Show Context)
Citation Context ...upward planar embedding in the process (whenever possible). This process can be extended to a drawing algorithm by transforming the embedding for use by the known algorithm of DiBattista and Tamassia =-=[10]-=- to draw single-source, single-sink planar DAGs. The main issue in this extension is the size of the output of the drawing algorithm, which we will address in Chapter 3 and again in Chapter 7. Upward ... |

56 |
Convex maps
- Stein
- 1951
(Show Context)
Citation Context ...ions. 68Q20, 68Q25, 68R05, 68R10 1. Introduction. There are a wide range of results dealing with drawing, representing, or testing planarity of graphs. Steinitz and Rademacher [22], F'ary [10], Stein =-=[21]-=-, and Wagner [26] independently showed that every planar graph can be drawn in the plane using only straight line segments for the edges. Tutte [25] showed that every 3-connected planar graph admits a... |

46 |
Small sets supporting Fáry embeddings of planar graphs
- Fraysseix, Pach, et al.
- 1988
(Show Context)
Citation Context ...ptimal, however we CHAPTER 3. RELATED RESULTS 16 shall see in Section 3.4 that for upward planar graphs, we will (provably) have this problem with any drawing algorithm. DeFraysseix, Pach and Pollack =-=[8]-=- have described an O(n log n) algorithm which takes an upward planar embedding (such as that from either planarity testing algorithm described) as input, and constructs a straight-line physical plane ... |

46 | Tollis, Algorithms for drawing graphs: an annotated bibliography, Computational Geometry 4 - Battista, Eades, et al. - 1994 |

44 |
Area requirement and symmetry display of planar upward drawings
- Battista, Tamassia, et al.
- 1992
(Show Context)
Citation Context ... induced subgraph of G on P (v). For 2 It is important to specify the time in arithmetic steps, because the algorithm is necessarily output sensitive: coordinates can require\Omega\Gamma n) bits each =-=[8]-=-. UPWARD PLANAR DRAWING OF SINGLE SOURCE ACYCLIC DIGRAPHS 5 a planar representation \Gamma of G, define \Gamma v to be the planar representation induced by \Gamma on G v . Proposition 3.2. Given a sin... |

38 |
Lattice Theory,” 3rd
- BIRKHOFF
- 1967
(Show Context)
Citation Context ...n, and greatest lower bound or meet. It is known that any planar poset with a least and greatest element is a lattice [21]. The theory of lattices is more developed than that of arbitrary posets (see =-=[2]), which m-=-ay be cause for algorithms for upward planarity testing of lattices to be "easier" than the more general case. Although any directed graph corresponds to some partially ordered set P , the m... |

37 | A linear-time algorithm for drawing planar graphs
- Chrobak, Payne
- 1995
(Show Context)
Citation Context ... straight lines between corresponding vertices, such that no two edges intersect except at their endpoints). This upper bound on the area of plane graph drawings applies to all planar graphs. Chrobak =-=[7]-=- (referenced by Tamassia and Eades' bibliography [29]) has improved their algorithm to linear time. Independently, the same result (upper bound and linear time) was achieved by Schnyder [27], with the... |

35 |
Linear algorithms for convex drawings of planar graphs
- Chiba, Yamanouchi, et al.
- 1984
(Show Context)
Citation Context ... on convex graphs, Thomassen [31] has presented inductive criteria for the existence of a convex representation of a graph in the plane. This was implemented as a linear time algorithm by Chiba et al =-=[5]. These ar-=-e significant to the topic at hand because they provide much of the graph-theoretical background evident in the works which followed. For general graphs, the problem of a "good" presentation... |

25 | Planar embedding of planar graphs
- Dolev, Leighton, et al.
- 1984
(Show Context)
Citation Context ...ould be interesting to characterize some classes of digraphs which permit straight-line upward plane drawings on a polynomially sized grid. Guaranteeing minimum area in all cases is, however, NP-hard =-=[9]-=-. The more general problem, testing upward planarity of an arbitrary acyclic digraph, has recently been shown to be NP-complete [12]. Another recent development by Bertolazzi and Di Battista [1] shows... |

25 |
Tutte, Convex Representations of Graphs
- T
- 1960
(Show Context)
Citation Context ...hs. Steinitz and Rademacher [22], F'ary [10], Stein [21], and Wagner [26] independently showed that every planar graph can be drawn in the plane using only straight line segments for the edges. Tutte =-=[25]-=- showed that every 3-connected planar graph admits a convex straight-line drawing, where the facial cycles other than the unbounded face are all convex polygons. The first linear time algorithm for te... |

21 | Planar lattices - Kelly, Rival - 1975 |

20 |
Fundamentals of planar ordered sets
- Kelly
- 1987
(Show Context)
Citation Context ...ane drawing is given. The digraph on the right is not upward planar---though it is planar, since placing v inside the face f would eliminate crossings, at the cost of producing a downward edge. Kelly =-=[17]-=- and Kelly and Rival [18], and v f Upward planar Non-upward-planar Fig. 1. Upward planar and non-upward planar digraphs. also Di Battista and Tamassia [7], have shown that for every upward plane drawi... |

16 |
I.: Bipartite graphs, upward drawings, and planarity
- Battista, Liu, et al.
- 1990
(Show Context)
Citation Context ...anar. An algorithm to find an upward plane drawing of such a digraph was given by Di Battista and Tamassia [7]. For the special case of bipartite digraphs, upward planarity is equivalent to planarity =-=[6]-=-. In this paper we will give an efficient algorithm to test upward planarity for single-source digraphs, eliminating the single-sink restriction. For the most part we will be concerned only with const... |

12 | On upward drawing testing of triconnected digraphs - Bertolazzi, Battista - 1991 |

11 |
Planar lattices and planar graphs
- Platt
- 1976
(Show Context)
Citation Context ...g upward planar digraphs has recently been shown to be NP-complete [12]. For the case of single-source single-sink digraphs there is a polynomial time recognition algorithm provided by Platt's result =-=[19]-=- that such a digraph is upward planar iff the digraph with a source-to-sink edge added is planar. An algorithm to find an upward plane drawing of such a digraph was given by Di Battista and Tamassia [... |

11 |
Planar acyclic oriented graphs
- Thomassen
- 1989
(Show Context)
Citation Context ...t. An upward plane drawing of a directed acyclic graph is a plane drawing of the digraph in which each directed edge is represented as a curve monotone increasing in the vertical direction. Thomassen =-=[24]-=- has given a non-algorithmic, graph-theoretic characterization of those directed graphs with a single source that admit an upward plane drawing. We present an efficient algorithm to test whether a giv... |

10 |
On determining the genus of a graph
- Filotti, Miller, et al.
- 1979
(Show Context)
Citation Context ...ble otherwise. If (u; v) is an edge of a digraph then u dominates v, u is incident to v, and v is incident from u. 3. A Combinatorial View of Upward Planarity. As discussed by Edmonds and others (see =-=[11]-=-) a connected graph G is planar iff it has a planar representation: a cyclic ordering of edges around each vertex such that the resulting set of faces F satisfies 2 = jF j \Gamma jEj + jV j (Euler's f... |

9 | Area requirement and symmetry display in drawing graphs - DiBattista, Tamassia, et al. - 1989 |

8 | Algorithms for drawing graphs: an annotated bibliography. Brown University TR - Tamassia, Eades - 1989 |

6 |
On straight line representations of planar graphs
- ary
- 1948
(Show Context)
Citation Context ... classifications. 68Q20, 68Q25, 68R05, 68R10 1. Introduction. There are a wide range of results dealing with drawing, representing, or testing planarity of graphs. Steinitz and Rademacher [22], F'ary =-=[10]-=-, Stein [21], and Wagner [26] independently showed that every planar graph can be drawn in the plane using only straight line segments for the edges. Tutte [25] showed that every 3-connected planar gr... |

6 | E cient Planarity Testing - Hopcroft, Tarjan - 1974 |

3 |
Graph Theory with Applications. MacMillian Co
- Bondy, Murty
- 1976
(Show Context)
Citation Context ...drawing from the abstract embedding of Chapter 7 (i.e. giving Euclidean coordinates for the vertices). Chapter 2 Preliminaries For the most part we use the terminology and notation of Bondy and Murty =-=[3]-=-, supplemented by that of Thomassen [30]. A (combinatorial) graph G consists of a set of vertices or nodes V and a set E of pairs of vertices called edges connecting these vertices. Late lower case Ro... |

2 |
Vorlesungen uber die Theorie de Polyeder
- Steinitz, Rademacher
- 1934
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Citation Context ... AMS subject classifications. 68Q20, 68Q25, 68R05, 68R10 1. Introduction. There are a wide range of results dealing with drawing, representing, or testing planarity of graphs. Steinitz and Rademacher =-=[22]-=-, F'ary [10], Stein [21], and Wagner [26] independently showed that every planar graph can be drawn in the plane using only straight line segments for the edges. Tutte [25] showed that every 3-connect... |

1 |
Bipartite graphs, upward drawings, and planarity. Universit`a Degli Studi di Roma, "La Sapienze
- DiBattista, Liu, et al.
- 1990
(Show Context)
Citation Context ... linear algorithm for this problem is given by DiBattista and Nardelli [11]. A very recent result of DiBattista, Liu and Rival is that any bipartite graph is upward-planar if and only if it is planar =-=[9]-=-. A graph G = (V; E) is bipartite if V can be partitioned into two disjoint subsets V 1 and V 2 such that every edge (u; v) of E connects u in V 1 (V 2 ) to v in (V 1 ) (i.e. no two vertices of V 1 ar... |

1 | An algorithm for testing planarity of hierarichical graphs - DiBattista, Nardelli |