  Using connected dominating set (CDS) to serve as a virtual backbone in a
wireless networks can save energy and reduce interference. Since nodes may fail
due to accidental damage or energy depletion, it is desirable that the virtual
backbone has some fault-tolerance. A $k$-connected $m$-fold dominating set
($(k,m)$-CDS) of a graph $G$ is a node set $D$ such that every node in
$V\setminus D$ has at least $m$ neighbors in $D$ and the subgraph of $G$
induced by $D$ is $k$-connected. Using $(k,m)$-CDS can tolerate the failure of
$\min\{k-1,m-1\}$ nodes. In this paper, we study Minimum Weight $(1,m)$-CDS
problem ($(1,m)$-MWCDS), and present an
$(H(\delta+m)+2H(\delta-1))$-approximation algorithm, where $\delta$ is the
maximum degree of the graph and $H(\cdot)$ is the Harmonic number. Notice that
there is a $1.35\ln n$-approximation algorithm for the $(1,1)$-MWCDS problem,
where $n$ is the number of nodes in the graph. Though our constant in $O(\ln
\cdot)$ is larger than 1.35, $n$ is replaced by $\delta$. Such a replacement
enables us to obtain a $(6.67+\varepsilon)$-approximation for the $(1,m)$-MWCDS
problem on unit disk graphs.
