## Which crossing number is it, anyway (1998)

Venue: | Proceedings of the 39th Annual Symposium on Foundations of Computer Science |

Citations: | 42 - 8 self |

### BibTeX

@INPROCEEDINGS{Pach98whichcrossing,

author = {János Pach and Géza Tóth},

title = {Which crossing number is it, anyway},

booktitle = {Proceedings of the 39th Annual Symposium on Foundations of Computer Science},

year = {1998},

pages = {617--626}

}

### Years of Citing Articles

### OpenURL

### Abstract

A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the minimum number of crossing points in any drawing of G. We define two new parameters, as follows. The pairwise crossing number (resp. the odd-crossing number) of G is the minimum number of pairs of edges that cross (resp. cross an odd number of times) over all drawings of G. We prove that the largest of these numbers (the crossing number) cannot exceed twice the square of the smallest (the odd-crossing number). Our proof is based on the following generalization of an old result of Hanani, which is of independent interest. Let G be a graph and let E0 be a subset of its edges such that there is a drawing of G, in which every edge belonging to E0 crosses any other edge an even number of times. Then G can be redrawn so that the elements of E0 are not involved in any crossing. Finally, we show that the determination of each of these parameters is an NP-hard problem and it is NP-complete in the case of the crossing number and the odd-crossing number. 1

### Citations

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Citation Context ...e same for the pair-crossing number. (See Remark at the end of Section 4.) 2 Proofs of Theorems 1 and 2 First we establish Theorem 1. The proof somewhat resembles a proof of Kuratowski's theorem (see =-=[BM76]-=-). Suppose that Theorem 1 is false. Then there exists a graph G with vertex set V (G) = V and edge set E(G) = E, and there is a subset E 0 # E such that G has a drawing, in which every edge in E 0 is ... |

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Citation Context ...| 2 - 1, which implies that # uv#E (|#(u) - #(v)| # K, as desired. # With Lemma 4.2, the proof of Theorem 4 (ii) is complete, because the Optimal Linear Arrangement Problem is known to be NP-complete =-=[GJS76]-=-. Remark. We can prove that the Pair Crossing Number Problem, pair-cr(G) # K, is NPhard. The proof is analogous to the proofs of the corresponding results for the crossing number (see [GJ83]) and for ... |

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Citation Context ...t in a proper drawing no two edges cross more than once, and if two edges share an endpoint, they cannot have another point in common ([WB78], [B91]). Many authors do not make this assumption ([T70], =-=[GJ83]-=-, [SSSV97]). If two edges are allowed to cross several times, we may count their intersections with multiplicity or without. We may also wish to impose some further restrictions on the drawings (e.g.,... |

133 | A Framework for Solving VLSI Graph Layout Problems
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Citation Context ... case of the crossing number and the odd-crossing number. 1 Introduction The crossing number of a graph G is usually defined as "the minimum number of edge crossings in any drawing of G in the pl=-=ane" [BL84]-=-. However, one has to be careful with this definition, because it can be interpreted in several ways. Sometimes it is assumed that in a proper drawing no two edges cross more than once, and if two edg... |

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Citation Context ...called even if it crosses every other edge an even number of times. It follows from Theorem A that if all edges of G are even, i.e., if odd-cr(G) = 0, then cr(G) = 0. (In this case, by Fary's theorem =-=[F48]-=-, we also have lin-cr(G) = 0.) In the next section, we establish the following generalization of this statement. Theorem 1. For a fixed drawing of a graph G, let G 0 # G denote the subgraph formed by ... |

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Citation Context ...bounded, then its crossing number cannot be arbitrarily large. More precisely, we prove Theorem 2. The crossing number of any graph G satisfies cr(G) # 2 (odd-cr(G)) 2 . It was discovered by Leighton =-=[L84]-=- that the crossing number can be used to obtain a lower bound on the chip area required for the VLSI circuit layout of a graph. For this purpose, he proved the following general lower bound for cr(G),... |

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Citation Context ...med that in a proper drawing no two edges cross more than once, and if two edges share an endpoint, they cannot have another point in common ([WB78], [B91]). Many authors do not make this assumption (=-=[T70]-=-, [GJ83], [SSSV97]). If two edges are allowed to cross several times, we may count their intersections with multiplicity or without. We may also wish to impose some further restrictions on the drawing... |

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Citation Context ...owing general lower bound for cr(G), which was discovered independently by Ajtai, Chvatal, Newborn, and Szemeredi. The best known constant, 1/33.75, in the theorem is due to Pach and Toth. Theorem B. =-=[ACNS82]-=-, [L84], [PT97] Let G be a graph with vertex set V (G) and edge set E(G) such that |E(G)| # 7.5|V (G)|. Then we have cr(G) # 1 33.75 |E(G)| 3 |V (G)| 2 . In Section 3, we prove that a similar inequali... |

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Citation Context ...e out the possibility that odd-cr(G) = pair-cr(G) = cr(G) for every graph G. The only result in this direction is the following remarkable theorem of Hanani and Tutte (see also [LPS97]). 2 Theorem A. =-=[Ch34]-=-, [T70] If a graph G can be drawn in the plane so that any two edges which do not share an endpoint cross an even number of times, then G is planar. For a generalization of this result to other surfac... |

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Citation Context ...nd for the complete bipartite graph K n,n with 2n vertices, as n tends to infinity [RT97]. The latter question, raised more than fifty years ago, is often referred to as Turan's Brick Factory Problem =-=[T77]-=- or as Zarankiewicz's problem [G69]. In the present paper, we investigate the relationship between various crossing numbers. First we agree on the terminology. A drawing of a simple undirected graph i... |

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Citation Context ...ward 667339. + Supported by OTKA-T-020914, OTKA-F-22234, and the Margaret and Herman Sokol Posdoctoral Fellowship Award. 1 must be straight-line segments [J71], or polygonal paths of length at most k =-=[BD93]-=-). No matter what definition we use, the determination of the crossing number of a graph appears to be an extremely di#cult task ([GJ83], [B91]). In fact, we do not even know the asymptotic value of a... |

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Citation Context ... K n,n with 2n vertices, as n tends to infinity [RT97]. The latter question, raised more than fifty years ago, is often referred to as Turan's Brick Factory Problem [T77] or as Zarankiewicz's problem =-=[G69]-=-. In the present paper, we investigate the relationship between various crossing numbers. First we agree on the terminology. A drawing of a simple undirected graph is a mapping f that assigns to each ... |

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Citation Context ... interpreted in several ways. Sometimes it is assumed that in a proper drawing no two edges cross more than once, and if two edges share an endpoint, they cannot have another point in common ([WB78], =-=[B91]-=-). Many authors do not make this assumption ([T70], [GJ83], [SSSV97]). If two edges are allowed to cross several times, we may count their intersections with multiplicity or without. We may also wish ... |

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Citation Context ...her hand, we cannot rule out the possibility that odd-cr(G) = pair-cr(G) = cr(G) for every graph G. The only result in this direction is the following remarkable theorem of Hanani and Tutte (see also =-=[LPS97]-=-). 2 Theorem A. [Ch34], [T70] If a graph G can be drawn in the plane so that any two edges which do not share an endpoint cross an even number of times, then G is planar. For a generalization of this ... |

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Citation Context ...er bound for cr(G), which was discovered independently by Ajtai, Chvatal, Newborn, and Szemeredi. The best known constant, 1/33.75, in the theorem is due to Pach and Toth. Theorem B. [ACNS82], [L84], =-=[PT97]-=- Let G be a graph with vertex set V (G) and edge set E(G) such that |E(G)| # 7.5|V (G)|. Then we have cr(G) # 1 33.75 |E(G)| 3 |V (G)| 2 . In Section 3, we prove that a similar inequality holds for th... |

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Citation Context ... we do not even know the asymptotic value of any of the above quantities for the complete graph K n with n vertices and for the complete bipartite graph K n,n with 2n vertices, as n tends to infinity =-=[RT97]-=-. The latter question, raised more than fifty years ago, is often referred to as Turan's Brick Factory Problem [T77] or as Zarankiewicz's problem [G69]. In the present paper, we investigate the relati... |

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Citation Context ...NSF grant CCR-94-24398 and PSC-CUNY Research Award 667339. + Supported by OTKA-T-020914, OTKA-F-22234, and the Margaret and Herman Sokol Posdoctoral Fellowship Award. 1 must be straight-line segments =-=[J71]-=-, or polygonal paths of length at most k [BD93]). No matter what definition we use, the determination of the crossing number of a graph appears to be an extremely di#cult task ([GJ83], [B91]). In fact... |

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Citation Context ...t can be interpreted in several ways. Sometimes it is assumed that in a proper drawing no two edges cross more than once, and if two edges share an endpoint, they cannot have another point in common (=-=[WB78]-=-, [B91]). Many authors do not make this assumption ([T70], [GJ83], [SSSV97]). If two edges are allowed to cross several times, we may count their intersections with multiplicity or without. We may als... |

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6 |
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Citation Context ...roper drawing no two edges cross more than once, and if two edges share an endpoint, they cannot have another point in common ([WB78], [B91]). Many authors do not make this assumption ([T70], [GJ83], =-=[SSSV97]-=-). If two edges are allowed to cross several times, we may count their intersections with multiplicity or without. We may also wish to impose some further restrictions on the drawings (e.g., the edges... |

1 |
Bounds for generalized thrackles, (manuscript
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(Show Context)
Citation Context ...If a graph G can be drawn in the plane so that any two edges which do not share an endpoint cross an even number of times, then G is planar. For a generalization of this result to other surfaces, see =-=[CN99]-=-. In a fixed drawing of a graph G, an edge is called even if it crosses every other edge an even number of times. It follows from Theorem A that if all edges of G are even, i.e., if odd-cr(G) = 0, the... |

1 |
Bounds for rectilinear crossing numbers, J. Graph Theory 17
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Citation Context ...ward 667339. † Supported by OTKA-T-020914, OTKA-F-22234, and the Margaret and Herman Sokol Posdoctoral Fellowship Award. 1must be straight-line segments [J71], or polygonal paths of length at most k =-=[BD93]-=-). No matter what definition we use, the determination of the crossing number of a graph appears to be an extremely difficult task ([GJ83], [B91]). In fact, we do not even know the asymptotic value of... |