We study the problem of finding the smallest power of an input graph that has <em>k</em> disjoint dominating sets, where the <em>i</em>th power of an input graph <em>G</em> is constructed by adding edges between pairs of vertices in <em>G</em> at distance <em>i</em> or less, and a subset of vertices in a graph <em>G</em> is a dominating set if and only if every vertex in <em>G</em> is adjacent to a vertex in this subset.   The problem is a different view of the <em>d</em>-domatic number problem in which the goal is to find the maximum number of disjoint dominating sets in the <em>d</em>th power of the input graph.   This problem is motivated by applications in multi-facility location and distributed networks. In the facility location framework, for instance, there are <em>k</em> types of services that all clients in different regions of a city should receive. A graph representing the map of regions in the city is given where the nodes of the graph represent regions and neighboring regions are connected by edges. The problem is how to establish facility servers in the city (each region can host at most one server) such that every client in the city can access a facility server in its region or in a region in the neighborhood. Since it may not be possible to find a facility location satisfying this condition, "a region in the neighborhood" required in the question is modified to "a region at the minimum possible distance <em>d</em>".   In this thesis, we study the connection of the above-mentioned problem with similar problems including the domatic number problem and the <em>d</em>-domatic number problem. We show that the problem is NP-complete for any fixed <em>k</em> greater than two even when the input graph is restricted to split graphs, <em>2</em>-connected graphs, or planar bipartite graphs of degree four. In addition, the problem is in P for bounded tree-width graphs, when considering <em>k</em> as a constant, and for strongly chordal graphs, for any <em>k</em>. Then, we provide a slightly simpler proof for a known upper bound for the problem. We also develop an exact (exponential) algorithm for the problem, running in time <em>O</em>(2. 73^<em>n</em>). Moreover, we prove that the problem cannot be approximated within ratio smaller than <em>2</em> even for split graphs, <em>2</em>-connected graphs, and planar bipartite graphs of degree four. We propose a greedy <em>3</em>-approximation algorithm for the problem in the general case, and other approximation ratios for permutation graphs, distance-hereditary graphs, cocomparability graphs, dually chordal graphs, and chordal graphs. Finally, we list some directions for future work.

Wireless Sensor Networks (WSNs) are now widely used for monitoring and controlling of systems where human intervention is not desirable or possible. Connected Dominating Sets (CDSs) based topology control in WSNs is one kind of hierarchical method to ensure sufficient coverage while reducing redundant connections in a relatively crowded network. Moreover, Minimum-sized Connected Dominating Set (MCDS) has become a well-known approach for constructing a Virtual Backbone (VB) to alleviate the broadcasting storm for efficient routing in WSNs extensively. However, no work considers the load-balance factor of CDSsin WSNs. In this dissertation, we first propose a new concept — the Load-Balanced CDS (LBCDS) and a new problem — the Load-Balanced Allocate Dominatee (LBAD) problem. Consequently, we propose a two-phase method to solve LBCDS and LBAD one by one and a one-phase Genetic Algorithm (GA) to solve the problems simultaneously. Secondly, since there is no performance ratio analysis in previously mentioned work, three problems are investigated and analyzed later. To be specific, the MinMax Degree Maximal Independent Set (MDMIS) problem, the Load-Balanced Virtual Backbone (LBVB) problem, and the MinMax Valid-Degree non Backbone node Allocation (MVBA) problem. Approximation algorithms and comprehensive theoretical analysis of the approximation factors are presented in the dissertation. On the other hand, in the current related literature, networks are deterministic where two nodes are assumed either connected or disconnected. In most real applications, however, there are many intermittently connected wireless links called lossy links, which only provide probabilistic connectivity. For WSNs with lossy links, we propose a Stochastic Network Model (SNM). Under this model, we measure the quality of CDSs using CDS reliability. In this dissertation, we construct an MCDS while its reliability is above a preset applicationspecified threshold, called Reliable MCDS (RMCDS). We propose a novel Genetic Algorithm (GA) with immigrant schemes called RMCDS-GA to solve the RMCDS problem. Finally, we apply the constructed LBCDS to a practical application under the realistic SNM model, namely data aggregation. To be specific, a new problem, Load-Balanced Data Aggregation Tree (LBDAT), is introduced finally. Our simulation results show that the proposed algorithms outperform the existing state-of-the-art approaches significantly.

We study the problem of finding the smallest power of an input graph that has <em>k</em> disjoint dominating sets, where the <em>i</em>th power of an input graph <em>G</em> is constructed by adding edges between pairs of vertices in <em>G</em> at distance <em>i</em> or less, and a subset of vertices in a graph <em>G</em> is a dominating set if and only if every vertex in <em>G</em> is adjacent to a vertex in this subset.   The problem is a different view of the <em>d</em>-domatic number problem in which the goal is to find the maximum number of disjoint dominating sets in the <em>d</em>th power of the input graph.   This problem is motivated by applications in multi-facility location and distributed networks. In the facility location framework, for instance, there are <em>k</em> types of services that all clients in different regions of a city should receive. A graph representing the map of regions in the city is given where the nodes of the graph represent regions and neighboring regions are connected by edges. The problem is how to establish facility servers in the city (each region can host at most one server) such that every client in the city can access a facility server in its region or in a region in the neighborhood. Since it may not be possible to find a facility location satisfying this condition, "a region in the neighborhood" required in the question is modified to "a region at the minimum possible distance <em>d</em>".   In this thesis, we study the connection of the above-mentioned problem with similar problems including the domatic number problem and the <em>d</em>-domatic number problem. We show that the problem is NP-complete for any fixed <em>k</em> greater than two even when the input graph is restricted to split graphs, <em>2</em>-connected graphs, or planar bipartite graphs of degree four. In addition, the problem is in P for bounded tree-width graphs, when considering <em>k</em> as a constant, and for strongly chordal graphs, for any <em>k</em>. Then, we provide a slightly simpler proof for a known upper bound for the problem. We also develop an exact (exponential) algorithm for the problem, running in time <em>O</em>(2. 73^<em>n</em>). Moreover, we prove that the problem cannot be approximated within ratio smaller than <em>2</em> even for split graphs, <em>2</em>-connected graphs, and planar bipartite graphs of degree four. We propose a greedy <em>3</em>-approximation algorithm for the problem in the general case, and other approximation ratios for permutation graphs, distance-hereditary graphs, cocomparability graphs, dually chordal graphs, and chordal graphs. Finally, we list some directions for future work.

A mobile ad hoc network consists of certain nodes that communicate only through wireless medium and can move arbitrarily. The key feature of a mobile ad hoc network is the mobility of the nodes. Because of the mobility, communication links form and disappear as nodes come into and go out of each other's communica- tion range. Mobile ad hoc networks are particularly useful in situations like disaster recovery and search, military operations, etc. Research on mobile ad hoc networks has drawn a huge amount of attention recently. The main challenges for mobile ad hoc networks are the sparse resources and frequent mobility. Most of the research work has been focused on the MAC and routing layer. In this work, we focus on distributed services for mobile ad hoc networks. These services will provide some fundamental functions in developing various applications for mobile ad hoc networks. In particular, we focus on the clock synchronization, connected dominating set, and k-mutual exclusion problems in mobile ad hoc networks.

A connected dominating set D is a set of vertices of a graph G=(V,E) such that every vertex in V-D is adjacent to at least one vertex in D and the subgraph induced by the set D is connected. The connected domination number is the minimum of the cardinalities of the connected dominating sets of G. The problem of finding a minimum connected dominating set D is known to be NP-hard. Many polynomial time algorithms that achieve some approximation factors have been provided earlier in finding a minimum connected dominating set. In this work, we present a survey on known properties of graph domination as well as some approximation algorithms. We implemented some of these algorithms and tested them with random graphs and compared their performance in finding a minimum connected dominating set D.

Online social networks (OSNs) have become a powerful medium of communicating, sharing and disseminating information. Because of popularity and availability of OSNs throughout the world, the connected users can spread information faster and thus propagate influence over each other constantly. Due to such impact, a lot of applications on OSNs focused on picking an initial set of users (seeds) to infuse their message in the OSN. Due to huge size of the network, the main challenge in picking the initial set is to maximize the resultant influence over the users in the network. The optimization problem of finding out the most influential set of members in an OSN for maximization of influence is an NP-hard problem. In this paper, we propose using the Positive Influential Dominating Set (PIDS) algorithm for the initial seed. PIDS is a well-known algorithm which determines the influential backbone nodes in the networks. We implemented PIDS-based influence maximization by using different propagation models. We compared PIDS performance to that of the existing approaches based on greedy and random heuristics. The experimental results from extensive simulation on real-world network data sets show that PIDS gives better influence spread than greedy and random for both Independent Cascade Model and Linear Threshold Model of influence propagation. PIDS is also scalable to large networks and in all size ranges, it performs well in influence maximization.

Wireless Sensor Networks (WSN) offer a flexible low-cost solution to the problem of event monitoring, especially in places with limited accessibility or that represent danger to humans. WSNs are made of resource-constrained wireless devices, which require energy efficient mechanisms, algorithms and protocols. One of these mechanisms is Topology Control (TC) composed of two mechanisms, Topology Construction and Topology Maintenance. This dissertation expands the knowledge of TC in many ways. First, it introduces a comprehensive taxonomy for topology construction and maintenance algorithms for the first time. Second, it includes four new topology construction protocols: A3, A3Lite, A3Cov and A3LiteCov. These protocols reduce the number of active nodes by building a Connected Dominating Set (CDS) and then turning off unnecessary nodes. The A3 and A3-Lite protocols guarantee a connected reduced structure in a very energy efficient manner. The A3Cov and A3LiteCov protocols are extensions of their predecessors that increase the sensing coverage of the network. All these protocols are distributed -they do not require localization information, and present low message and computational complexity. Third, this dissertation also includes and evaluates the performance of four topology maintenance protocols: Recreation (DGTRec), Rotation (SGTRot), Rotation and Recreation (HGTRotRec), and Dynamic Local-DSR (DLDSR). Finally, an event-driven simulation tool named Atarraya was developed for teaching, researching and evaluating topology control protocols, which fills a need in the area of topology control that other simulators cannot. Atarraya was used to implement all the topology construction and maintenance cited, and to evaluate their performance. The results show that A3Lite produces a similar number of active nodes when compared to A3, while spending less energy due to its lower message complexity. A3Cov and A3CovLite show better or similar coverage than the other distributed protocols discussed here, while preserving the connectivity and energy efficiency from A3 and A3Lite. In terms of network lifetime, depending on the scenarios, it is shown that there can be a substantial increase in the network lifetime of 450% when a topology construction method is applied, and of 3200% when both topology construction and maintenance are applied, compared to the case where no topology control is used.

Consider a configuration of pebbles distributed on the vertices of a connected graph of order n. A pebbling step consists of removing two pebbles from a given vertex and placing one pebble on an adjacent vertex. A distribution of pebbles on a graph is called solvable if it is possible to place a pebble on any given vertex using a sequence of pebbling steps. The pebbling number of a graph, denoted f (G), is the minimal number of pebbles such that every configuration of f (G) pebbles on G is solvable. We derive several general upper bounds on the pebbling number, improving previous results.

For a graph G = (V, E), a set S ⊆ V is a total dominating set if every vertex in V is adjacent to some vertex in S. The smallest cardinality of any total dominating set is the total domination number γt(G). For an arbitrary edge e εE(Ḡ), γt(G) - 2 ≤ γt(G + e) ≤ γt(G); if the latter inequality is strict for each e ε E(Ḡ) ≠ φ, then G is said to be γt-critical. The criticality index of an edge e ε E(Ḡ) is γt(e) = γt(G) - γt(G + e). Let E(Ḡ) = [e1...,em} and S = ∑j=1m̄ci(ej). The criticality index of G is ci(G) = S/m̄. For any rational number k, 0 ≤ k ≤ 2, we construct a graph G with ci(G) = k. For 1 ≤ k ≤ 2, we construct graphs with this property that are γt-critical as well as graphs that are not γt-critical.

A set S of vertices in a graph G = (V, E) is a locating-dominating set if S is a dominating set of G, and every pair of distinct vertices {u, v} in V - S is located with respect to S, that is, if the set of neighbors of u that are in S is not equal to the set of neighbors of v that are in S. We give a construction of trees that have unique minimum locating-dominating sets.