  To save energy and alleviate interferences in a wireless sensor network, the
usage of virtual backbone was proposed. Because of accidental damages or energy
depletion, it is desirable to construct a fault tolerant virtual backbone,
which can be modeled as a $k$-connected $m$-fold dominating set (abbreviated as
$(k,m)$-CDS) in a graph. A node set $C\subseteq V(G)$ is a $(k,m)$-CDS of graph
$G$ if every node in $V(G)\backslash C$ is adjacent with at least $m$ nodes in
$C$ and the subgraph of $G$ induced by $C$ is $k$-connected. In this paper, we
present an approximation algorithm for the minimum $(3,m)$-CDS problem with
$m\geq3$. The performance ratio is at most $\gamma$, where
$\gamma=\alpha+8+2\ln(2\alpha-6)$ for $\alpha\geq4$ and $\gamma=3\alpha+2\ln2$
for $\alpha<4$, and $\alpha$ is the performance ratio for the minimum
$(2,m)$-CDS problem. Using currently best known value of $\alpha$, the
performance ratio is $\ln\delta+o(\ln\delta)$, where $\delta$ is the maximum
degree of the graph, which is asymptotically best possible in view of the
non-approximability of the problem. This is the first performance-guaranteed
algorithm for the minimum $(3,m)$-CDS problem on a general graph. Furthermore,
applying our algorithm on a unit disk graph which models a homogeneous wireless
sensor network, the performance ratio is less than 27, improving previous ratio
62.3 by a large amount for the $(3,m)$-CDS problem on a unit disk graph.
