Generalized Domination in Graphs with Applications in Wireless Networks

The objective of this research is to study practical generalization of domination in graphs and explore the theoretical and computational aspects of models arising in the design of wireless networks. For the construction of a virtual backbone of a wireless ad-hoc network, two different models are proposed concerning reliability and robustness. This dissertation also considers wireless sensor placement problems with various additional constraints that reflect different real-life scenarios. In wireless ad-hoc network, a connected dominating set (CDS) can be used to serve as a virtual backbone, facilitating communication among the members in the network. Most literature focuses on creating the smallest virtual backbone without considering the distance that a message has to travel from a source until it reaches its desired destination. However, recent research shows that the chance of loss of a message in transmission increases as the distance that the message has to travel increases. We propose CDS with bounded diameter, called dominating s-club (DsC) for s ≥ 1, to model a reliable virtual backbone. An ideal virtual backbone should retain its structure after the failure of a certain number of vertices. The issue of robustness under vertex failure is considered by studying k-connected m-dominating set. We describe several structural properties that hold form ≥ k, but fail when m < k. Three different formulations based on vertex-cut inequalities are shown depending on the value of k and m. The computational results show that the proposed lazy-constraint approach compares favorably with existing methods for the minimum connected dominating set problem (for k = m = 1). The experimental results for k = m = 2, 3, 4 are presented as well. In the wireless sensor placement problem, the objective is often to place a minimum number of sensors while monitoring all sites of interest, and this can be modeled by dominating set. In some practical situations, however, there could be a location where a sensor cannot be placed because of environmental restrictions. Motivated by these practical scenarios, we introduce varieties of dominating set and the corresponding optimization problems. For these new problems, we consider the computational complexity, mathematical programming formulation, and analytical bounds on the size of structures of interest. We solve these problems using a general commercial solver and compare its performance with that of simulated annealing

Efficient Data Dissemination in Wireless Ad Hoc Networks

In this thesis, we study the problem of efficient data dissemination in wireless sensor and mobile ad hoc networks. In wireless sensor networks we study two problems: (1) construction of virtual backbones and clustering hierarchies to achieve efficient routing, and (2) placement of multiple sinks, where each sensor is at a bounded distance to several sinks, to analyze and process data before sending it to a central unit. Often connected dominating sets have been used for such purposes. However, a connected dominating set is often vulnerable due to frequent node failures in wireless sensor networks. Hence, to provide a degree of fault-tolerance we consider in problem (1) a 2-connected (k,r)-dominating set, denoted D(2,k,r), to act as a virtual backbone or a clustering hierarchy, and in problem (2) a total (k,r)-dominating set to act as sinks in wireless sensor networks. Ideally, the backbone or the number of sinks in the network should constitute the smallest percentage of nodes in the network. We model the wireless sensor network as a graph. The total (k,r)-dominating set and the 2-connected (k,r)-dominating set have not been studied in the literature. Thus, we propose two centralized approximation algorithms to construct a D(2,k,r) in unit disk graphs and in general graphs. We also derive upper bounds on the total (k,r)-domination number in graphs of girth at least 2k+1 as well as in random graphs with non-fixed probability p. In mobile ad hoc networks we propose a hexagonal based beacon-less flooding algorithm, HBLF, to efficiently flood the network. We give sufficient condition that even in the presence of holes in the network, HBLF achieves full delivery. Lower and upper bounds are given on the number of forwarding nodes returned by HBLF in a network with or without holes. When there are no holes in the network, we show that the ratio of the shortest path returned by HBLF to the shortest path in the network is constant. We also present upper bounds on the broadcast time of HBLF in a network with or without holes