On the Longest Edge of Gabriel Graphs in Wireless Ad Hoc Networks

In wireless ad hoc networks, without fixed infrastructures, virtual backbones are constructed and maintained to efficiently operate such networks. The Gabriel graph (GG) is one of widely used geometric structures for topology control in wireless ad hoc networks. If all nodes have the same maximal transmission radii, the length of the longest edge of the GG is the critical transmission radius such that the GG can be constructed by localized and distributed algorithms using only 1-hop neighbor information. In this paper, we assume a wireless ad hoc network is represented by a Poisson point process with mean n on a unit-area disk, and nodes have the same maximal transmission radii. We give three asymptotic results on the length of the longest edge of the GG. First, we show that the ratio of the length of the longest edge to radic(ln n/pin)is asymptotically almost surely equal to 2. Next, we show that for any xi, the expected number of GG edges whose lengths are at least 2radic(ln n + xi/pin) is asymptotically equal to 2e
^middotxi
. This implies that xi rarr chi is an asymptotically almost sure sufficient condition for constructing the GG by 1-hop information. Last, we prove that the number of long edges is asymptotically Poisson with mean 2e
^-xi
. Therefore, the probability of the event that the length of the longest edge is less than 2radic(ln n + xi/pin) is asymptotically equal to exp(-2e
^-xi
)