Random triangulated surfaces are created by taking an even number, n, of triangles and arbitrarily ”gluing” together pairs of edges until every edge has been paired. The resulting surface can be described in terms of its number of boundary cycles, a random variable denoted by h. Building upon the work of Nicholas Pippenger and Kristin Schleich, and using a recent result from Alex Gamburd, we establish an improved approximation for the expectation of h for certain values of n. We use a computer simulation to exactly determine the distribution of h for small values of n, and present a method for calculating these probabilities. We also conduct an investigation into the related problem of creating one connected component out of n triangles.

Wireless Sensor Networks (WSNs) comprise a large number of sensor nodes, which are spread out within a region and communicate using wireless links. In some WSN applications, recognizing boundary nodes is important for topology discovery, geographic routing and tracking. In this thesis, we study the problem of identifying the boundary nodes of a WSN. In a WSN, close-by nodes can communicate with their neighbors and have the ability to estimate distances to nearby nodes, but not necessarily the true distances. Our objective is to find the boundary nodes by using the connectivity relation and neighbor distance information without any knowledge of node locations. Moreover, our main aim is to design a distributed algorithm that works even when the average degree is low. We propose a heuristic algorithm to find the boundary nodes which are connected in a boundary cycle of a location-free, low density, randomly deployed WSN. We develop the key ideas of our boundary detection algorithm in the centralized scenario and extend these ideas to the distributed scenario. Then, we show by simulation experiments that the distributed implementation outperforms the centralized one. The centralized implementation relies on the connectivity of the network to the base station. Therefore, for low density disconnected networks, the algorithm cannot find boundaries in partitions of the network that cannot establish connection to the base station. This condition leads to a low quality of boundary discovery. In contrast, the distributed implementation is more realistic for real WSNs, especially for relatively sparse networks when all local information cannot be collected very well due to sparse connectivity. In low-degree disconnected networks, the simulation results show that the distributed implementation has a higher quality of boundaries compared to the centralized implementation.