## Finding shortest non-separating and non-contractible cycles for topologically embedded graphs (2005)

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Venue: | Discrete Comput. Geom |

Citations: | 39 - 8 self |

### BibTeX

@INPROCEEDINGS{Cabello05findingshortest,

author = {Sergio Cabello and Bojan Mohar},

title = {Finding shortest non-separating and non-contractible cycles for topologically embedded graphs},

booktitle = {Discrete Comput. Geom},

year = {2005},

pages = {131--142},

publisher = {Springer-Verlag}

}

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### Abstract

We present an algorithm for finding shortest surface non-separating cycles in graphs embedded on surfaces in O(g 3/2 V 3/2 log V + g 5/2 V 1/2) time, where V is the number of vertices in the graph and g is the genus of the surface. If g = o(V 1/3−ε), this represents a considerable improvement over previous results by Thomassen, and Erickson and Har-Peled. We also give algorithms to find a shortest non-contractible cycle in O(g O(g) V 3/2) time, which improves previous results for fixed genus. This result can be applied for computing the (non-separating) face-width of embedded graphs. Using similar ideas we provide the first near-linear running time algorithm for computing the face-width of a graph embedded on the projective plane, and an algorithm to find the face-width of embedded toroidal graphs in O(V 5/4 log V) time. 1

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Citation Context ...gical disk, and GP is a planar graph. In GP, each cycle from Qx corresponds to two paths on the boundary of D. We can glue an infinite number of copies of D to construct the universal cover of Σ; see =-=[13]-=- for a reference on universal covers. However, because of Lemma 8, we can find a shortest non-contractible cycle by constructing only a portion of this universal cover. These are the main ideas to pro... |

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Citation Context ...ree T such that for any vertex u we have dG(v,u) = dT(v,u). Since E = O(V +g), a shortest-path tree from any given vertex can be computed in O(V log V + E) = O(V log V + g) time using Fibonacci heaps =-=[12]-=-. When g = O(V 1−ε ) for any positive, fixed ε, then a shortest path tree can be constructed in O(V ) time 1 . In the special case that all the edge-lengths are equal to one, any breadth-first-search ... |

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Citation Context ...e cycle ˜ C of length O( � 2V/(2g) log 2g) if V ≥ g, and O(log 2g) if g > V . The projection C of ˜ C to T is a closed walk in T which is noncontractible because of the homotopy lifting property (cf. =-=[19]-=-). This walk contains a noncontractible cycle of length at most |V ( ˜ C)| = ew(DT), so ew(T) ≤ ew(DT), which is what we were to prove. � 7.2 Face-width in the projective plane For the special case wh... |

200 |
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Citation Context ...t quantifies local planarity and density of embeddings. The face-width is closely related to the edge-width, the minimum number of vertices of any shortest non-contractible cycle of an embedded graph =-=[20]-=-. Among some relevant applications, face-width plays a fundamental role in the graph minors theory of Robertson and Seymour, and large face-width implies that there exists a collection of cycles that ... |

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Citation Context ...ti) for i = 1,... ,k. 1 Eppstein [8] shows how to compute in linear time a separator S of size O( √ gV ) = O(V 1−ε/2 ) for G such that G − S is planar. The recursive subdivision that Henzinger et al. =-=[14]-=- require can then be obtained using the division by Eppstein in the first level and then continue in each planar subpiece using their approach 5sDjidjev [4] and Fakcharoenphol and Rao [11] (slightly i... |

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Citation Context ...optimal system of loops and represent it implicitly in O((V + g)log V ) time; an explicit representation may need Θ(gV ) space. Other known results for curves embedded on topological surfaces include =-=[2, 3, 18, 24]-=-; see also [21, 22] and references therein. 2 Background We describe the topological and graph-theoretical background assumed through the paper. 3sTopology. We consider surfaces Σ that are connected, ... |

55 | Exact and Approximate Distances in Graphs - A Survey
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Citation Context ... of computation, that is, we only add and compare (sums of) edge weights. For integer weights and word-RAM model of computation, some logarithmic improvements may be possible. See the survey by Zwick =-=[25]-=- for a discussion. 3 k-pairs distance problem Consider the k-pairs distance problem: Given a graph G with positive edge-weights and k pairs (s1,t1),... ,(sk,tk) of vertices of G, compute the distances... |

53 | Planar graphs, negative weight edges, shortest paths, and near linear time
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Citation Context ...nger et al. [14] require can then be obtained using the division by Eppstein in the first level and then continue in each planar subpiece using their approach 5sDjidjev [4] and Fakcharoenphol and Rao =-=[11]-=- (slightly improved by Klein [17] for nonnegative edge-lengths) describe data structures for shortest path queries in planar graphs. We will need the following special case. Lemma 1 For a planar graph... |

48 | Optimally cutting a surface into a disk
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Citation Context ...t cycle in QQ for Q ∈ Qx. Firstly, we use that Tx is a shortest-path tree to argue that we only need to consider cycles that intersect Q exactly once; a similar idea is used by Erickson and Har-Peled =-=[9]-=- for their 2-approximation algorithm. Secondly, we reduce the problem of finding a shortest cycle in QQ to an O(V )-pairs distance problem. We describe the whole algorithm in pseudocode at the end of ... |

45 |
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Citation Context ... be obtained using the division by Eppstein in the first level and then continue in each planar subpiece using their approach 5sDjidjev [4] and Fakcharoenphol and Rao [11] (slightly improved by Klein =-=[17]-=- for nonnegative edge-lengths) describe data structures for shortest path queries in planar graphs. We will need the following special case. Lemma 1 For a planar graph of order V , the k-pairs distanc... |

40 |
Embeddings of graphs with no short noncontractible cycles
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Citation Context ...In Section 3 we describe 2sresults on the k-pairs distance problem that we use later on. The rest of the sections are as described above; we conclude in Section 8. 1.2 Related previous work Thomassen =-=[23]-=- was the first to give a polynomial time algorithm for finding a shortest nonseparating and a shortest non-contractible cycle in a graph on a surface; see also [20, Chapter 4]. Although Thomassen does... |

40 |
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Citation Context ...optimal system of loops and represent it implicitly in O((V + g)log V ) time; an explicit representation may need Θ(gV ) space. Other known results for curves embedded on topological surfaces include =-=[2, 3, 18, 24]-=-; see also [21, 22] and references therein. 2 Background We describe the topological and graph-theoretical background assumed through the paper. 3sTopology. We consider surfaces Σ that are connected, ... |

35 |
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Citation Context ...optimal system of loops and represent it implicitly in O((V + g)log V ) time; an explicit representation may need Θ(gV ) space. Other known results for curves embedded on topological surfaces include =-=[2, 3, 18, 24]-=-; see also [21, 22] and references therein. 2 Background We describe the topological and graph-theoretical background assumed through the paper. 3sTopology. We consider surfaces Σ that are connected, ... |

35 | Dynamic Generators of Topologically Embedded Graphs
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Citation Context ...tion on the edge-lengths, they provide polynomial time algorithms for both problems. The same authors have also given polynomial time algorithms for finding optimal pants decompositions [6]. Eppstein =-=[8]-=- discusses how to use the tree-cotree partition for dynamically maintaining properties from a graph under several operations. For example, he can maintain the minimum and maximum spanning tree under e... |

19 | Transforming curves on surfaces
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(Show Context)
Citation Context |

10 |
On the independence ratio of a graph
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(Show Context)
Citation Context ...is a continuous function of the circle S 1 in Σ. Two loops are homotopic if there is a continuous deformation of one onto the other, that is, if there is a continuous function from the cylinder S 1 × =-=[0,1]-=- to Σ such that each boundary of the cylinder is mapped to one of the loops. A loop is contractible if it is homotopic to a constant (a loop whose image is a single point); otherwise it is non-contrac... |

9 |
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(Show Context)
Citation Context ...nd represent it implicitly in O((V + g)log V ) time; an explicit representation may need Θ(gV ) space. Other known results for curves embedded on topological surfaces include [2, 3, 18, 24]; see also =-=[21, 22]-=- and references therein. 2 Background We describe the topological and graph-theoretical background assumed through the paper. 3sTopology. We consider surfaces Σ that are connected, compact, Hausdorff ... |

7 |
On short noncontractible cycles in embedded graphs
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(Show Context)
Citation Context ...nd face-width of G can be computed in O(g O(g) V 3/2 ) time. It can happen that ew(G) = Ω(V ). The situation is different for the face-width fw(G) for which there exist non-trivial bounds. Hutchinson =-=[15]-=- showed that the edge-width of a triangulation in an orientable surface of genus g ≤ V is O( � V/g log g), and O(log g) if g > V , improving the previous bound of √ 2V by Albertson and Hutchinson [1].... |

6 |
On-Line Algorithms for Shortest Path Problems on Planar Digraphs
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(Show Context)
Citation Context ...ecursive subdivision that Henzinger et al. [14] require can then be obtained using the division by Eppstein in the first level and then continue in each planar subpiece using their approach 5sDjidjev =-=[4]-=- and Fakcharoenphol and Rao [11] (slightly improved by Klein [17] for nonnegative edge-lengths) describe data structures for shortest path queries in planar graphs. We will need the following special ... |

3 |
de Verdière. Shortening of curves and decomposition of surfaces. PhD thesis, Université Paris 7, December 2003. Avilable at http://www.di.ens.fr/users/colin/ textes/these.html.en. É. Colin de Verdière and
- Colin
- 2004
(Show Context)
Citation Context ...ontractible) cycle whose length is at most twice the length of a shortest one. Several other algorithmical problems for graphs embedded on surfaces have been considered. Colin de Verdière and Lazarus =-=[7]-=- considered the problem of finding a shortest cycle in a given homotopy class, as well as a system of loops homotopic to a given one. Under some realistic assumption on the edge-lengths, they provide ... |

1 | An algorithm for embedding graphs in the torus
- Juvan, Mohar
(Show Context)
Citation Context ...( √ V ). Using the previous lemma with t = O( √ V ), we conclude the following. Theorem 14 Let G be a graph embedded in P 2 . We can compute the face-width of G in O(V log 2 V ) time. Juvan and Mohar =-=[16]-=- obtained a linear time algorithm for deciding if fw(G) ≤ k, where k is a fixed constant. They needed the special case when k = 4 in an algorithm for testing embeddability in the torus. 7.3 Face-width... |

1 |
Paths in graphs and curves on surfaces
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(Show Context)
Citation Context ...nd represent it implicitly in O((V + g)log V ) time; an explicit representation may need Θ(gV ) space. Other known results for curves embedded on topological surfaces include [2, 3, 18, 24]; see also =-=[21, 22]-=- and references therein. 2 Background We describe the topological and graph-theoretical background assumed through the paper. 3sTopology. We consider surfaces Σ that are connected, compact, Hausdorff ... |