Independent and dominating sets in wireless communication graphs

Wireless ad hoc networks are advancing rapidly, both in research and more and more into our everyday lives. Wireless sensor networks are a prime example of a new technology that has gained a lot of attention in the literature, and that is going to enhance the way we view and interact with the environment. These ad hoc and sensor networks are modeled by communication graphs which give the communication links between some devices or nodes equipped with wireless transceivers or radios. The nature of wireless transmissions does not lead to an arbitrary network topology, but creates a network with certain properties. These properties include the important structure of bounded growth. In this thesis, we look at the structures and resulting optimization problems of independent and dominating sets in on graphs that model wireless communication networks. Independent and dominating sets are prominently used in the efficient organization of large-scale wireless ad hoc and sensor networks. In a communication graph, an independent set consists of vertices that cannot communicate with one another directly. Such a set is commonly used in clustering strategies, e.g. to obtain a hierarchical view of the network. A dominating set is given by a set of vertices so that every vertex in the graph is either in this set, or adjacent to a vertex from this set. Dominating sets of small cardinality are frequently used for backbone structures in communication networks, e.g. to obtain efficient multi-hop routing protocols. For the optimization problems of seeking independent sets of large cardinality or weight (Maximum Independent Set problem), and seeking dominating sets of small cardinality (Minimum Dominating Set problem) on graphs of polynomially bounded growth, we present and discuss polynomial-time approximation schemes (PTAS). The algorithms presented are robust in the sense that they accept an undirected graph, and return a desired solution or a certificate that shows that the instance does not satisfy the structural properties of a wireless