## Abstract

The cover time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of ad-hoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have been the subject of much study. A random geometric graph G(n, r) is obtained by placing n points uniformly at random on the unit square and connecting two points iff their Euclidean distance is at most r. The phase transition behavior with respect to the radius r of such graphs has been of special interest. We show that there exists a critical radius r_{opt} such that for any r ≥ r _{opt} G(n, r) has optimal cover time of Θ(r_{con}) with high probability, and, importantly, r_{opt} = Θ(r_{COn}) where r_{COn} denotes the critical radius guaranteeing asymptotic connectivity. Moreover, since a disconnected graph has infinite cover time, there is a phase transition and the corresponding threshold width is O(r _{con}). We are able to draw our results by giving a tight bound on the electrical resistance of G(n, r) via the power of certain constructed flows.

Original language | English |
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Pages (from-to) | 677-689 |

Number of pages | 13 |

Journal | Lecture Notes in Computer Science |

Volume | 3580 |

DOIs | |

State | Published - 1 Jan 2005 |

Externally published | Yes |

Event | 32nd International Colloquium on Automata, Languages and Programming, ICALP 2005 - Lisbon, Portugal Duration: 11 Jul 2005 → 15 Jul 2005 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science (all)