## Computing the Girth of a Planar Graph (2000)

Venue: | In Proc. 27th International Colloquium on Automata, Languages and Programming ICALP 2000, volume 1853 of LNCS |

Citations: | 5 - 0 self |

### BibTeX

@INPROCEEDINGS{Djidjev00computingthe,

author = {Hristo N. Djidjev},

title = {Computing the Girth of a Planar Graph},

booktitle = {In Proc. 27th International Colloquium on Automata, Languages and Programming ICALP 2000, volume 1853 of LNCS},

year = {2000},

pages = {821--831},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

The girth of a graph G has been de ned as the length of a shortest cycle of G. We design an O(n 5=4 log n) algorithm for finding the girth of an undirected n-vertex planar graph, giving the first o(n 2 ) algorithm for this problem. Our approach combines several techniques such as graph separation, hammock decomposition, covering of a planar graph with graphs of small tree-width, and dynamic shortest path computation. We discuss extensions and generalizations of our result.

### Citations

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Citation Context ...and generalizations of our result. 1 Introduction Given an (unweighted) graph G, the length of a path p in G is the number of the edges in p. The girth of G (denoted by girth(G)) was dened by Harary [=-=15-=-] as the length of a shortest cycle in G, or innity if G has no cycle. The girth is a basic combinatorial characteristic of graphs and its relations to other graph properties have been extensively stu... |

736 |
Graph Theory
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Citation Context ...e of certain type of minors of the graph. Other results relate the girth of the graph to the minimum or the average degrees of its vertices, its diameter, its maximum genus, and its connectivity (see =-=[6]-=-). Thesrst ecient algorithm for computing the girth of graph was given by Itai and Rodeh [16], who describe an O(nm) algorithm for computing the girth of a general n-vertex m-edge graph G. They also d... |

403 | A separator theorem for planar graphs
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Citation Context ...dded in the plane is a plane graph. A separator of G is a set of vertices whose removal leaves no connected component of more than n=2 vertices. If G is planar, then G has a separator of size O( p n) =-=[18, 9]-=- and if G has genus g > 0, then G has a separator of O( p gn) vertices [8, 14]. In both cases, the corresponding separators can be found in O(n) time. If a graph G has non-negative weights associated ... |

312 |
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Citation Context ...ection we assume that the girth of the input graph is smaller than certain parameterswhose value will be determined in Section 5. For the proof of the next lemma we use a technique developed by Baker =-=[3]-=- and Eppstein [11]. Lemma 2. Let G be an n-vertex planar graph and let d be any integer. Then we cansnd in O(n) time a set of subgraphs G i of G with the following properties: (i) The sum of the sizes... |

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Citation Context ...oblem asks, given a source vertex s of G, to compute the distances between s and all other vertices of G. If G is planar, then the single-source shortest path problem for G can be solved in O(n) time =-=[17]-=-. The graph G is planar if G can be embedded in the plane so that no two edges intersect except at a common endpoint. A planar graph of n vertices has at most 3n 3 = O(n) edges. A graph already embedd... |

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(Show Context)
Citation Context ... cycle in G, or innity if G has no cycle. The girth is a basic combinatorial characteristic of graphs and its relations to other graph properties have been extensively studied. In particular, Erdos [1=-=2-=-], Lovasz [19], Bollobas [4], Cook [5], and others studied the relationship between the girth and the chromatic number of a graph. Thomassen [22] and Mader [20] studied the relationship between the gi... |

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Citation Context ... an O(n) algorithm forsnding a triangle in the graph, if one exists (and thus solves the girth problem for planar graphs in case girth(G) 3 in O(n) time). Their results were generalized by Eppstein [=-=11]-=-, who developed an O(n) algorithm forsnding the girth of a planar graph G provided girth(G) = O(1). (His algorithm, however, is superexponential with respect to girth(G).) ? This work was partially su... |

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Citation Context ... minimum or the average degrees of its vertices, its diameter, its maximum genus, and its connectivity (see [6]). Thesrst ecient algorithm for computing the girth of graph was given by Itai and Rodeh =-=[16]-=-, who describe an O(nm) algorithm for computing the girth of a general n-vertex m-edge graph G. They also design an O(n 2 ) algorithm for computing the girth within an additive error of one. Finding s... |

81 |
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Citation Context ...e removal leaves no connected component of more than n=2 vertices. If G is planar, then G has a separator of size O( p n) [18, 9] and if G has genus g > 0, then G has a separator of O( p gn) vertices =-=[8, 14]-=-. In both cases, the corresponding separators can be found in O(n) time. If a graph G has non-negative weights associated with its vertices, then a weighted separator of G is a set of vertices whose r... |

35 |
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Citation Context ...ts> 0. We will use the dynamic shortest path algorithm from [7] that runs faster for graphs of small face-on-vertex cover. A hammock decomposition of an n-vertex graph G was dened by Frederickson in [=-=13]-=- as a decomposition of G into certain outerplanar digraphs called hammocks. Hammocks satisfy the following properties: (i) each hammock has at most four vertices, called attachment vertices, shared wi... |

21 |
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(Show Context)
Citation Context ...dded in the plane is a plane graph. A separator of G is a set of vertices whose removal leaves no connected component of more than n=2 vertices. If G is planar, then G has a separator of size O( p n) =-=[18, 9]-=- and if G has genus g > 0, then G has a separator of O( p gn) vertices [8, 14]. In both cases, the corresponding separators can be found in O(n) time. If a graph G has non-negative weights associated ... |

18 |
Finding and counting given length cycles. Algorithmica
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Citation Context ...of even and odd lengths have been studied by Monien [21] and Vazirani and Yannakakis [23]. There are numerous results onsnding a cycle of a given length in general or special graphs; see e.g. [1] and =-=[2]-=- for recent results and references. In the case of planar graphs Itai and Rodeh [16] give an O(n) algorithm forsnding a triangle in the graph, if one exists (and thus solves the girth problem for plan... |

15 | Improved algorithms for dynamic shortest paths
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(Show Context)
Citation Context ...rected n-vertex planar graph in O(n 5=4 log n) time. Our approach makes use of recently developed fast dynamic algorithms for computing shortest paths in planar graphs with small face-on-vertex cover =-=[7]-=-. If G has an appropriately large girth, then we show that the shortest cycle in G can be computed by combining separator based divide-and-conquer with the dynamic shortest path algorithm from [7]. If... |

9 |
A separator theorem
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(Show Context)
Citation Context ...e removal leaves no connected component of more than n=2 vertices. If G is planar, then G has a separator of size O( p n) [18, 9] and if G has genus g > 0, then G has a separator of O( p gn) vertices =-=[8, 14]-=-. In both cases, the corresponding separators can be found in O(n) time. If a graph G has non-negative weights associated with its vertices, then a weighted separator of G is a set of vertices whose r... |

7 |
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(Show Context)
Citation Context ...ral n-vertex m-edge graph G. They also design an O(n 2 ) algorithm for computing the girth within an additive error of one. Finding shortest cycles of even and odd lengths have been studied by Monien =-=[21]-=- and Vazirani and Yannakakis [23]. There are numerous results onsnding a cycle of a given length in general or special graphs; see e.g. [1] and [2] for recent results and references. In the case of pl... |

4 |
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(Show Context)
Citation Context ...le. The girth is a basic combinatorial characteristic of graphs and its relations to other graph properties have been extensively studied. In particular, Erdos [12], Lovasz [19], Bollobas [4], Cook [5=-=]-=-, and others studied the relationship between the girth and the chromatic number of a graph. Thomassen [22] and Mader [20] studied the relationship between the girth and the existence of certain type ... |

4 |
On chromatic number of set-systems
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Citation Context ...or innity if G has no cycle. The girth is a basic combinatorial characteristic of graphs and its relations to other graph properties have been extensively studied. In particular, Erdos [12], Lovasz [1=-=9-=-], Bollobas [4], Cook [5], and others studied the relationship between the girth and the chromatic number of a graph. Thomassen [22] and Mader [20] studied the relationship between the girth and the e... |

3 |
A linear algorithm for partitioning graphs of genus. Serdica : Bulgaricae mathematicae publicationes
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Citation Context ...hermore, using graph separation and divide-and-conquer, we can compute the girth of a directed graph in O(n 3=2 ) time. Finally, making use of the O( p gn) separator theorem for graphs of genus g > 0 =-=[8, 14, 10]-=-, one can construct ecient algorithms for the above problems for graphs of genus g = o(n). We will describe algorithms for solving the above problems in the full version of the paper. This paper leave... |

2 |
Chromatic number, girth and maximal degree. Discrete Mathematics
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(Show Context)
Citation Context ...has no cycle. The girth is a basic combinatorial characteristic of graphs and its relations to other graph properties have been extensively studied. In particular, Erdos [12], Lovasz [19], Bollobas [4=-=]-=-, Cook [5], and others studied the relationship between the girth and the chromatic number of a graph. Thomassen [22] and Mader [20] studied the relationship between the girth and the existence of cer... |

1 |
Topological subgraphs in graphs of large girth
- Mader
- 1998
(Show Context)
Citation Context ...ensively studied. In particular, Erdos [12], Lovasz [19], Bollobas [4], Cook [5], and others studied the relationship between the girth and the chromatic number of a graph. Thomassen [22] and Mader [2=-=0]-=- studied the relationship between the girth and the existence of certain type of minors of the graph. Other results relate the girth of the graph to the minimum or the average degrees of its vertices,... |

1 |
circuits and subdivisions
- Paths
(Show Context)
Citation Context ...s have been extensively studied. In particular, Erdos [12], Lovasz [19], Bollobas [4], Cook [5], and others studied the relationship between the girth and the chromatic number of a graph. Thomassen [2=-=2]-=- and Mader [20] studied the relationship between the girth and the existence of certain type of minors of the graph. Other results relate the girth of the graph to the minimum or the average degrees o... |

1 |
orientations, 0-1 permanents, and even cycles in directed graphs
- Pfaan
- 1989
(Show Context)
Citation Context ... also design an O(n 2 ) algorithm for computing the girth within an additive error of one. Finding shortest cycles of even and odd lengths have been studied by Monien [21] and Vazirani and Yannakakis =-=[23]-=-. There are numerous results onsnding a cycle of a given length in general or special graphs; see e.g. [1] and [2] for recent results and references. In the case of planar graphs Itai and Rodeh [16] g... |