## Degree constrained book embeddings (2002)

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Venue: | J. Algorithms |

Citations: | 2 - 1 self |

### BibTeX

@ARTICLE{Wood02degreeconstrained,

author = {David R. Wood},

title = {Degree constrained book embeddings},

journal = {J. Algorithms},

year = {2002},

volume = {45},

pages = {144--154}

}

### OpenURL

### Abstract

A book embedding of a graph consists of a linear ordering of the vertices along a line in 3-space (the spine), and an assignment of edges to half-planes with the spine as boundary (the pages), so that edges assigned to the same page can be drawn on that page without crossings. Given a graph G = (V, E), letf: V → N be a function such that 1 � f(v)�deg(v). We present a Las Vegas algorithm which produces a book embedding of G with O ( � |E|·maxv⌈deg(v)/f (v) ⌉ ) pages, such that at most f(v)edges incident to avertexvareonasingle page. This result generalises that of Malitz [J. Algorithms 17 (1)

### Citations

1869 | Randomized Algorithms - Motwani, Raghavan - 1995 |

253 |
A decomposition theorem for partially ordered sets
- Dilworth
- 1950
(Show Context)
Citation Context .... Two edges are incomparable under � if and only if they intersect. Thus an antichain is a completely intersecting set of edges, and a chain is a set of pairwise disjoint edges. By Dilworth’s Theorem =-=[5]-=- there is a decomposition of E into k chains where k is the size of the largest antichain. That is, there is a k-page degree-1 book embedding of G with spine ordering π. The time complexity can be ach... |

114 |
The complexity of coloring circular arcs and chords
- Garey, Johnson, et al.
- 1980
(Show Context)
Citation Context ...w) �= ρ(xy).Thebook thickness or page number of agraphG is the minimum number of pages in a book embedding of G. Determining the book thickness of a graph is NP-hard, even with a fixed spine ordering =-=[11]-=-. A number of results establish upper bounds on the book thickness of certain classes of graphs [1,6,7,10,18], such as the celebrated theorem of Yannakakis [23] that every planar graph has book thickn... |

83 |
The book thickness of a graph
- Bernhart, Kainen
- 1979
(Show Context)
Citation Context ...a graph G = (V, E) by n =|V |, the number of edges of G by m =|E|, and the maximum degree of G by ∆(G), or∆ if the graph in question is clear. Book embeddings, first introduced by Bernhart and Kainen =-=[1]-=-, are a graph layout style with numerous applications (see [4]). A book consists of a line in 3-space, called the spine, and a number of pages, each a half-plane with the spine as boundary. A book emb... |

48 | Embedding graphs in books: a layout problem with applications to VLSI design
- Chung, Leighton, et al.
- 1987
(Show Context)
Citation Context ...|E|, and the maximum degree of G by ∆(G), or∆ if the graph in question is clear. Book embeddings, first introduced by Bernhart and Kainen [1], are a graph layout style with numerous applications (see =-=[4]-=-). A book consists of a line in 3-space, called the spine, and a number of pages, each a half-plane with the spine as boundary. A book embedding (π, ρ) of a graph consists of a linear ordering π of th... |

43 | Laying out graphs using queues - HEATH, ROSENBERG - 1992 |

25 |
Genus g graphs have pagenumber O
- Malitz
- 1994
(Show Context)
Citation Context ... book thickness of certain classes of graphs [1,6,7,10,18], such as the celebrated theorem of Yannakakis [23] that every planar graph has book thickness at most four. For graphs with genus γ , Malitz =-=[14]-=- proved that the book thickness is O( √ γ).Since γ � m, the book thickness is O( √ m), a result proved independently by the same author [15]. While the proofs of Malitz are probabilistic, Shahrokhi an... |

16 |
Graphs with E edges have pagenumber O
- Malitz
- 1994
(Show Context)
Citation Context ...ok thickness at most four. For graphs with genus γ , Malitz [14] proved that the book thickness is O( √ γ).Since γ � m, the book thickness is O( √ m), a result proved independently by the same author =-=[15]-=-. While the proofs of Malitz are probabilistic, Shahrokhi and Shi [20] describe a deterministic algorithm, which given a vertex k-colouring of a graph G, computes a book embedding of G with O( √ km) p... |

15 |
On nontrivial separators for k-page graphs, and simulations by nondeterministic one-tape Turing machines
- Galil, Kannan, et al.
(Show Context)
Citation Context ...in which the page degree of every vertex v is at most f(v). If for all vertices v ∈ V , f(v)= c for some constant c,a degree-f book embedding is simply called a degree-c book embedding. Galil, et al. =-=[8,9]-=- refer to a graph which admits a degree-1 book embedding with k pages as a k-pushdown graph. Motivated by problems in computational complexity, they established lower bounds on the size of a separator... |

13 | Bounds for orthogonal 3-D graph drawing - Biedl, Shermer, et al. - 1999 |

11 |
On the pagenumber of complete bipartite graphs
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- 1997
(Show Context)
Citation Context ...edding of G. Determining the book thickness of a graph is NP-hard, even with a fixed spine ordering [11]. A number of results establish upper bounds on the book thickness of certain classes of graphs =-=[1,6,7,10,18]-=-, such as the celebrated theorem of Yannakakis [23] that every planar graph has book thickness at most four. For graphs with genus γ , Malitz [14] proved that the book thickness is O( √ γ).Since γ � m... |

11 |
The pagenumber of k-trees is
- Ganley, Heath
(Show Context)
Citation Context ...edding of G. Determining the book thickness of a graph is NP-hard, even with a fixed spine ordering [11]. A number of results establish upper bounds on the book thickness of certain classes of graphs =-=[1,6,7,10,18]-=-, such as the celebrated theorem of Yannakakis [23] that every planar graph has book thickness at most four. For graphs with genus γ , Malitz [14] proved that the book thickness is O( √ γ).Since γ � m... |

11 | Pagenumber of Complete Bipartite Graphs - Muder, Weaver, et al. - 1988 |

8 |
Orthogonal 3-D graph drawing
- Biedl, Shermer, et al.
- 1999
(Show Context)
Citation Context ...h k pages as a k-pushdown graph. Motivated by problems in computational complexity, they established lower bounds on the size of a separator in 3pushdown graphs. Implicit in the work of Biedl, et al. =-=[2]-=- is a degree-1 book embedding of the complete graph Kn with n pages. In this paper we consider the following problem: given a graph G = (V, E) and an arbitrary constraint function f of G, produce a de... |

7 |
On 3-pushdown graphs with large separators
- GALIL, KANNAN, et al.
- 1989
(Show Context)
Citation Context ...in which the page degree of every vertex v is at most f(v). If for all vertices v ∈ V , f(v)= c for some constant c,a degree-f book embedding is simply called a degree-c book embedding. Galil, et al. =-=[8,9]-=- refer to a graph which admits a degree-1 book embedding with k pages as a k-pushdown graph. Motivated by problems in computational complexity, they established lower bounds on the size of a separator... |

6 |
The pagenumber of toroidal graphs is at most seven
- Endo
- 1997
(Show Context)
Citation Context ...edding of G. Determining the book thickness of a graph is NP-hard, even with a fixed spine ordering [11]. A number of results establish upper bounds on the book thickness of certain classes of graphs =-=[1,6,7,10,18]-=-, such as the celebrated theorem of Yannakakis [23] that every planar graph has book thickness at most four. For graphs with genus γ , Malitz [14] proved that the book thickness is O( √ γ).Since γ � m... |

6 |
On a generalization of edge-coloring in graphs
- Hakimi, Kariv
- 1986
(Show Context)
Citation Context ...number of edges incident to a vertex on a single page is constrained. (A similar approach is taken for the graph-theoretic thickness by Bose and Prabhu [3], and for edge colouring by Hakimi and Kariv =-=[12]-=-.) We define the page degree of a vertex v in a particular book embedding to be the maximum number of edges incident to v on a single page. A constraint function of a graph G = (V, E) is a function f ... |

5 | Algebraic Discrete Methods - SIAM - 1984 |

2 |
Thickness of graphs with degree constrained vertices
- Bose, Prabhu
- 1977
(Show Context)
Citation Context ...n this paper we study book embeddings where the number of edges incident to a vertex on a single page is constrained. (A similar approach is taken for the graph-theoretic thickness by Bose and Prabhu =-=[3]-=-, and for edge colouring by Hakimi and Kariv [12].) We define the page degree of a vertex v in a particular book embedding to be the maximum number of edges incident to v on a single page. A constrain... |

2 |
Optimal partition of a bipartite graph into non-crossing matchings
- Malucelli, Nicoloso
(Show Context)
Citation Context ...y can be achieved using a dual form of the algorithm by Heath and Rosenberg [13, Theorem 2.3]. ✷ Note that an equivalent result to Lemma 2 with a more lengthy proof is given by Malucelli and Nicoloso =-=[16]-=-. To enable Lemma 2 to be extended to degree-f book embeddings, consider the following construction. Let π be a linear ordering of the vertices of a graph G = (V, E),andletf be a constraint function o... |

2 | Graphs with E edges have pagenumber O( p - Algorithms - 1994 |

1 | Genus g graphs have pagenumber O (v/-ff - Malitz - 1994 |

1 | The complexity of coloring circular arcs and chords - Hakimi, Kariv - 1980 |

1 | Laying out graphs using queues - Theory - 1986 |

1 | Genus g graphs have pagenumber O( p - Comput - 1992 |

1 | Optimal partition of a bipartite graph into non-crossing matchings - Algorithms - 1994 |