In a wireless ad hoc network, the size of the virtual backbone (VB) is an important factor for measuring the quality of the VB. The smaller the VB is, the less the overhead caused by the VB. Since ball graphs (BGs) have been used to model 3-dimensional wireless ad hoc networks and since a connected dominating set can be used to represent a VB undertaking routing-related tasks, the problem of finding the smallest VB is transformed into the problem of finding a minimum connected dominating set (MCDS). Many research results on the MCDS problem have been obtained for unit disk graphs and unit ball graphs, in which the transmission ranges of all nodes are identical. In some situations, the node powers can vary. One can model such a network as a graph with different transmission ranges for different nodes. In this paper, we focus on the problem of minimum strongly connected dominating and absorbing sets (MSCDASs) in a strongly connected directed ball graph with different transmission ranges, which is also NP-hard. We design an algorithm considering the construction of a strongly connected dominating and absorbing set (SCDAS), whose size does not exceed 319/15k3+116/5k2+29/5kopt+29/3k3+116/5k2+87/5k+13/15, where opt is the size of an MCDAS and k denotes the ratio of rmax to rmin in the ad hoc network with transmission range rmin,rmax. Our simulations show the feasibility of the algorithm proposed in this paper.

We usually use a digraph to represent a wireless network (WN). Correspondingly, a connected dominating set (CDS) of the digraph is usually used to denote a virtual backbone (VB) of the corresponding WN. In this article, focusing on the problem of a minimum strongly connected dominating and absorbing set (MSCDAS) with a bounded diameter (or guaranteed routing cost) for a digraph, which is strongly connected, we introduce two algorithms. One is called the guaranteed routing cost strongly connected dominating and absorbing set (GOC-SCDAS), which can generate a strongly connected dominating and absorbing set (SCDAS) with a performance ratio 14.4k+1/22 in respect of the optimal solution. Another is called the α guaranteed routing cost strongly connected bidirectional dominating and absorbing set (α-GOC-SCBDAS), which can generate a strongly connected bidirectional dominating and absorbing set (SCBDAS) with a performance ratio 8.8443k+1/22k+1/22 in respect of the optimal solution and a better routing cost, where k=rmax/rmin and rmin,rmax is the transmission range of nodes in the network. Through the simulation experiments, we obtain the conclusion that in terms of the diameter and average routing path length (ARPL) of CDS, the outputs of our algorithms are better than those of the algorithm in (Du et al. 2006).