## Undirected Vertex-Connectivity Structure and Smallest Four-Vertex-Connectivity Augmentation (1995)

Venue: | Proc. 6th ISAAC |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Hsu95undirectedvertex-connectivity,

author = {Tsan-sheng Hsu},

title = {Undirected Vertex-Connectivity Structure and Smallest Four-Vertex-Connectivity Augmentation},

booktitle = {Proc. 6th ISAAC},

year = {1995},

pages = {920}

}

### OpenURL

### Abstract

In this paper, we study properties for the structure of an undirected graph that is not 4-vertex-connected. We also study the evolution of this structure when an edge is added to optimally increase the vertex-connectivity of the underlying graph. Several properties reported here can be extended to the case of a graph that is not k-vertex- connected, for an arbitrary k. Using properties obtained here, we solve the problem of finding a smallest set of edges whose addition 4-vertex-connects an undirected graph. This is a fundamental problem in graph theory and has applications in network reliability and in statistical data security. We give an O(n \Delta log n + m)-time algorithm for finding a set of edges with the smallest cardinality whose addition 4-vertex-connects an undirected graph, where n and m are the number of vertices and edges in the input graph, respectively. This is the first polynomial time algorithm for this problem when the input graph is not 3-vertex-connecte...

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Citation Context ...lem that is studied here (see the survey chapter in [Hsu93]) and dynamic graph algorithms [LP91]. The structure of an undirected graph that is not biconnected (i.e., 2-vertex-connected) is well-known =-=[Har69]-=- and is represented as a 2-block graph. The structure of a biconnected graph that is not triconnected (i.e., 3-vertex-connected) is also well-known and is represented as a 3-block graph [HT73, Tut66].... |

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Citation Context ...mial time approximation algorithm that uses no more than k extra edges for augmenting a (k \Gamma 1)-vertex-connected directed graph to achieve k-vertex-connectivity. Very recently, Frank and Jord'an =-=[FJ93]-=- gave a polynomial time algorithm to solve the smallest vertex-connectivity augmentation problem on directed graphs exactly. Their algorithm increases the vertex-connectivity of a directed graph by an... |

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Citation Context ...or93b], and for four-connectivity [Hsu92]. However, it is difficult to see a general framework for the case of increasing the vertex-connectivity by more than one from the results in [HR91, WN93]. In =-=[HR91]-=-, a counter example is given to show that we cannot optimally raise the vertex-connectivity of a graph to three by first optimally raising the vertex-connectivity to two and then using the special alg... |

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Citation Context ...lso results known for augmenting planar graphs and outerplanar graphs [Kan93b]. The above results are for augmenting undirected graphs. For directed graph augmentation, Masuzawa, Hagihara, and Tokura =-=[MHT87]-=- studied this problem when the input graph is a directed oriented tree. Their algorithm runs in O( \Delta n) time wheresis the vertexconnectivity of the resulting graph. Jord'an [Jor93a] gave a polyno... |

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Citation Context ...ial algorithm for finding a smallest augmentation to triconnect a graph with n vertices and m edges. Hsu and Ramachandran [HR91] gave a linear-time algorithm for this problem. (Independently, Jord'an =-=[Jor93b]-=- gave a different linear-time algorithm for the special case of optimally triconnecting a biconnected graph.) Hsu [Hsu92] also gave an almost linear-time algorithm for fourconnecting a triconnected gr... |

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Citation Context ...ntial algorithm. An O(log 2 n)-time parallel algorithm on an EREW PRAM using a linear number of processors for this problem was also given in Hsu and Ramachandran [HR93]. Fern'andez-Baca and Williams =-=[FBW89]-=- considered the smallest augmentation problem for reaching biconnectivity on hierarchically defined graphs. This version of the augmentation problem has applications in VLSI circuit design. They obtai... |

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Citation Context ...vertex-connectivity by one is first reported in [ET76] for the case of reaching biconnectivity. This framework has been extended for reaching triconnectivity [HR91, Jor93b], and for four-connectivity =-=[Hsu92]-=-. However, it is difficult to see a general framework for the case of increasing the vertex-connectivity by more than one from the results in [HR91, WN93]. In [HR91], a counter example is given to sho... |

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Citation Context ...tion The following results are known for solving the smallest augmentation problem on an undirected graph to satisfy a given vertex-connectivity requirement. Eswaran and Tarjan [ET76] (and Plesn ' ik =-=[Ple76]-=-, independently) gave a lower bound for the smallest number of edges needed to biconnect an undirected graph and proved that the lower bound can always be achieved. Rosenthal and Goldner [RG77] develo... |

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