An important problem in wireless ad hoc and sensor networks is to select a few nodes to form a virtual backbone that supports routing and other tasks such as area monitoring. Previous work in this area has focused on selecting a small vir-tual backbone for high efficiency. In this paper, we propose the construction of a k-connected k-dominating set (k-CDS) as a backbone to balance efficiency and fault tolerance. Four localized k-CDS construction protocols are proposed. The first pro-tocol randomly selects virtual backbone nodes with a given probability pk, where pk depends on the value of k and network condition, such as network size and node density. The second one maintains a fixed backbone node degree of Bk, where Bk also depends on the network condition. The third protocol is a deterministic ap-proach. It extends Wu and Dai’s coverage condition, which is originally designed for 1-CDS construction, to ensure the formation of a k-CDS. The last protocol is a hybrid of probabilistic and deterministic approaches. It provides a generic frame-work that can convert many existing CDS algorithms into k-CDS algorithms. These protocols are evaluated via a simulation study. Key words: Connected dominating set (CDS), k-vertex connectivity, localized algorithms, simulation, wireless ad hoc and sensor networks. PACS: Preprint submitted to Elsevier Science 23 September 2005

Abstract- Ad hoc network normally has critical connectivity properties before partitioning. The timely recognition is important in order to perform some data or service replication. Several existing centralized or globalized algorithms declare an edge or a node as critical if their removal will separate the network into several components. We introduce several localized definitions of critical nodes and critical links, using topological or positional information. A node is critical if the subgraph of k-hop neighbours of node (without the node itself) is disconnected. We propose three definitions of critical links, based on verifying common k-hop neighbours, loop length, and critical status of link endpoints, respectively. The experiments with random unit graph model of ad hoc networks show high correspondence of local and global decisions. For instance, in experiments with 500 nodes in connected random unit graphs (using a novel graph generation method from [6], first published here), over half of locally estimated critical nodes and links were indeed globally critical even for k=1 (the accuracy increases to over 70 % for k=2 and over 80% for k=3), for average number of neighbours ranging from 3 to 15. The errors mostly occur when alternative routes exist but are relatively long, and therefore may not provide satisfactory service in applications. Therefore our localized protocols provide faster and often more reliable partition warnings for possible timely replication decisions. I.

Energy efficiency and network capacity are two of the most important issues in wireless sensor networks. Topology-control algorithms have been proposed to maintain network connectivity while reducing energy consumption and improving network capacity. Several studies have also been performed to investigate critical conditions on several network parameters in order to ensure network k-connectivity (in the asymptotic sense). In this chapter, several problems (and their corresponding solutions) related to topology construction, maintenance, and connectivity in wireless sensor networks are discussed. Specifically, topics discussed include (1) various communication models and generation of random network topologies; (2) neighbor discovery and maintenance; (3) basic connectivity properties of wireless sensor networks (with the random unit graph model as the underlying model); (4) localized topology construction algorithms, along with their associated geometric structures in both homogeneous and heterogeneous networks; and (5) how to enhance fault tolerance in topology construction and connectivity. 10.1