### Citations

1694 | Graph Theory with Applications.
- Bondy
- 1976
(Show Context)
Citation Context ... er sio ns1s-s92 Paul D. Manuel, Bharati Rajan, Indra Rajasingh, P. Vasanthi Beulah 2.3 Lower Bound We first solve the crossing number problem for BF3. We begin with the following results. Theorem 5 (=-=[3]-=-, Euler’s formula) In a connected plane graph G with n vertices, ε edges and f faces (regions), n− ε+ f = 2. Theorem 6 ([3]) If G is a connected plane graph with girth g, then gf ≤ 2ε. Lemma 2 Let G b... |

164 | A framework for solving vlsi graph layout problems
- Bhatt, Leighton
- 1984
(Show Context)
Citation Context ...ea layout as well as on the number of wire contact cuts that should be minimized. Discoveries of Leighton in the early 1980’s made this, an area of high importance in the theoretical computer science =-=[1, 6]-=-. The chip area required for the realization of an electrical network is closely related to the crossing number of the underlying graph. In information visualization, one of the most important measure... |

142 | Crossing numbers and hard Erdös problems in discrete geometry
- Székely
- 1993
(Show Context)
Citation Context ...inimum number of crossings of its edges among all drawings of G in the plane. It is an important measure of the non-planarity of a graph [12], with applications in discrete and computational geometry =-=[11]-=-. The crossing number of a graph corresponding to the VLSI circuit has strong influence on the area layout as well as on the number of wire contact cuts that should be minimized. Discoveries of Leight... |

105 |
Complexity Issues in VLSI
- Leighton
- 1983
(Show Context)
Citation Context ...ea layout as well as on the number of wire contact cuts that should be minimized. Discoveries of Leighton in the early 1980’s made this, an area of high importance in the theoretical computer science =-=[1, 6]-=-. The chip area required for the realization of an electrical network is closely related to the crossing number of the underlying graph. In information visualization, one of the most important measure... |

40 | A successful concept for measuring non-planarity of graphs: the crossing number
- Székely
(Show Context)
Citation Context ... 1 Introduction The crossing number cr(G) of a graph G is the minimum number of crossings of its edges among all drawings of G in the plane. It is an important measure of the non-planarity of a graph =-=[12]-=-, with applications in discrete and computational geometry [11]. The crossing number of a graph corresponding to the VLSI circuit has strong influence on the area layout as well as on the number of wi... |

33 |
An experimental study of the basis for graph drawing algorithms
- Purchase, Cohen, et al.
- 1997
(Show Context)
Citation Context ...etwork is closely related to the crossing number of the underlying graph. In information visualization, one of the most important measures of the quality of a graph drawing is the number of crossings =-=[9]-=-. The problem is NP-hard [5]. Cimikowski [4] has given an upper bound for the crossing number of the butterfly network as 3 2 4r +O(r2r) †Email: pauldmanuel@gmail.com. Supported by UNESCO-HP under ”Br... |

8 | On the Bisection Width and Expansion of Butterfly Networks, Theory of Computing Systems 34
- Bornstein, Litman, et al.
- 2001
(Show Context)
Citation Context ...ce (d1, d2 ... dn) and bisection width bw(G), cr(G) ≥ 1 40 bw2(G)− 1 16 ∑ d2v, (1) ha l-0 09 80 75 8,sv er sio ns1s-sImproved Bounds on the Crossing Number of Butterfly Network 93 It is also shown in =-=[2]-=- that for an 2r-input butterfly network, the bisection width is given by, bw(G) = 0.82× 2r (2) upto error terms. Combining the results (1) and (2), we get immediately that, cr(G) ≥ 1 40 bw2(G)− 1 16 ∑... |

4 |
An efficient representation of benes networks and its applications
- Manuel, Abd-El-Barra, et al.
- 2008
(Show Context)
Citation Context ...nce (DMTCS), Nancy, France ha l-0 09 80 75 8,sv er sio ns1s-s88 Paul D. Manuel, Bharati Rajan, Indra Rajasingh, P. Vasanthi Beulah where r denotes the dimension of the network. In 2008, Manuel et al. =-=[7]-=- announced a drawing (D) of this butterfly network for which we obtained [10] an improved bound as cr(BFr) ≤ 4r +O(r2r), r ≥ 3. In this paper we propose a new drawing (φ) of the butterfly network in t... |

3 | Topological Properties of Some Interconnection Network Graphs
- Cimikowski
- 1996
(Show Context)
Citation Context ...mber of the underlying graph. In information visualization, one of the most important measures of the quality of a graph drawing is the number of crossings [9]. The problem is NP-hard [5]. Cimikowski =-=[4]-=- has given an upper bound for the crossing number of the butterfly network as 3 2 4r +O(r2r) †Email: pauldmanuel@gmail.com. Supported by UNESCO-HP under ”Brain Gain Initiative” Scheme (Project Number ... |

3 |
Applications of the Crossing
- Pach, Shahrokhi, et al.
- 1996
(Show Context)
Citation Context ...order O(4r). A lower bound of matching order can be simply obtained by combining the known facts. Theorem 8 Let G be an r-dimensional butterfly network. Then cr(G) ≥ 1 59 4r − r2r + 2r−1. It is known =-=[8]-=- that for an n-vertex graph with the degree sequence (d1, d2 ... dn) and bisection width bw(G), cr(G) ≥ 1 40 bw2(G)− 1 16 ∑ d2v, (1) ha l-0 09 80 75 8,sv er sio ns1s-sImproved Bounds on the Crossing N... |

2 |
Crossing Number is NP
- Garey, Johnson
- 1983
(Show Context)
Citation Context ... the crossing number of the underlying graph. In information visualization, one of the most important measures of the quality of a graph drawing is the number of crossings [9]. The problem is NP-hard =-=[5]-=-. Cimikowski [4] has given an upper bound for the crossing number of the butterfly network as 3 2 4r +O(r2r) †Email: pauldmanuel@gmail.com. Supported by UNESCO-HP under ”Brain Gain Initiative” Scheme ... |

2 |
On the Crossing Number of Butterfly Networks
- Rajan, Rajasingh, et al.
- 2010
(Show Context)
Citation Context ...el, Bharati Rajan, Indra Rajasingh, P. Vasanthi Beulah where r denotes the dimension of the network. In 2008, Manuel et al. [7] announced a drawing (D) of this butterfly network for which we obtained =-=[10]-=- an improved bound as cr(BFr) ≤ 4r +O(r2r), r ≥ 3. In this paper we propose a new drawing (φ) of the butterfly network in the plane with 1 4 4r +O(r2r) number of crossings. We also give a lower bound ... |