In wireless sensor networks, virtual backbone has been proposed as the routing infrastructure to alleviate the broadcasting storm problem and perform some other tasks such as area monitoring. Previous work in this area has mainly focused on how to set up a small virtual backbone for high efficiency, which is modelled as the minimum Connected Dominating Set (CDS) problem. In this paper we consider how to establish a small virtual backbone to balance efficiency and fault tolerance. This problem can be formalized as the minimum
-dominating set problem, which is a general version of minimum CDS problem with
= 1 and
= 1. In this paper we will propose some approximation algorithms for this problem that beat the current best performance ratios.