Two fundamental functions of the sensor nodes in a wireless sensor network are to sense its environment and to transmit sensed information to a basestation. One approach to prolong sensor network lifetime is to deploy some relay nodes whose main function is to communicate with the sensor nodes, other relay nodes, and the basestations. It is desirable to deploy a minimum number of relay nodes to achieve certain connectivity requirement. In this paper, we study four related fault-tolerant relay node placement problems, each of which has been previously studied only in some restricted form. For each of them, we discuss its computational complexity and present a polynomial time O(1)-approximation algorithm with a small approximation ratio. When the problem reduces to a previously studied form, our algorithm either improves the previous best algorithm or reduces to the previous best algorithm.