k-Strongly Connected m-Dominating and Absorbing Set in Wireless Ad Hoc Networks with Unidirectional Links Ravi Tiwari Tania Mishra {rtiwari, tmishra}@cise.ufl.edu C.I.S.E Department University of Florida Gainesville, FL 32611 Yingshu Li C.S. Department, Georgia State University Atlanta, GA, 30303 yli@cs.gsu.edu My T. Thai C.I.S.E Department University of Florida Gainesville, FL 32611 mythai@cise.ufl.edu Abstract Since there is no fixed infrastructure or centralized management in wireless ad hoc networks (WANs), vir- tual backbone has been proposed as the routing infras- tructure to alleviate the broadcasting storm problem [6]. Because the virtual backbone nodes need to carry other node’s traffic and subject to failure, it is desirable to construct a fault tolerant virtual backbone. Most recent research has studied this problem in homogeneous net- works. In this paper, we propose solutions for efficient construction of a fault-tolerant virtual backbone where the wireless nodes have different transmission ranges. Such a network can be modeled as a disk graph where any link between two nodes is either unidirectional or bidirectional. Since the graph is directed, we formu- late the fault tolerant virtual backbone problem as a k- Strongly Connected m-Dominating and Absorbing Set problem (k-m-SCDAS). We first propose two heuristics, one to construct a 1-m-SCDAS and the other one to ob- tain a k-1-SCDAS in a directed graph. We next combine these two approaches to develop a general construction of k-m-SCDAS. Through extensive simulations, we com- pare the performance of these proposed algorithms. 1 Introduction In wireless ad hoc networks (WANs), there is no fixed or predefined infrastructure. Nodes in WANs commu- nicate via shared medium, either through a single hop communication or multi-hop relays. In order to enable data transfer in such networks, all the wireless nodes need to frequently flood control messages, thus caus- ing a lot of redundancy, contentions, and collisions [6]. As a result, a virtual backbone has been proposed as the routing infrastructure of a network for efficient rout- ing, broadcasting and collision avoidance protocols [14]. With virtual backbones, routing messages are only ex- changed between the backbone nodes, instead of being broad casted to all the nodes. With the help of virtual backbone, routing is easier and can adapt quickly to net- work topology changes. It has also shown that virtual backbones could dramatically reduce routing overhead [15]. Furthermore, using virtual backbone as forward- ing nodes can efficiently reduce the energy consump- tion, which is also one of the critical issue in WANs. Since the virtual backbone nodes need to carry other node’s traffic, fault tolerance must be considered. Un- fortunately, virtual backbone is often very vulnerable due to frequent node failure and link failure, which is inherent in wireless ad hoc networks. Therefore, con- structing a fault tolerant virtual backbone that continues to function during node or link failure is an important research problem, which has not studied sufficiently. In [5, 1], the authors considered this problem in Unit Disk Graph (UDG) [16], in which all nodes have the same transmission ranges. However, transmission ranges of all nodes in a net- work are not necessarily equal. Nodes in a network may have different power due to difference in functionali- ties, power control to alleviate collisions, topology con- trol to achieve a certain level of connectivity. For ex- ample, in heterogeneous networks, gateway nodes may have higher power than other nodes. Or in some topol- ogy controlled networks, each node may adjust its trans- mission range differently to obtain certain optimization goals. All these scenarios result into a WAN with dif- ferent transmission ranges. Such a network can be mod- eled as a Disk Graph (DG) G. Note that G is directed, 1