Rainbow connection number, bridges and radius

Let $G$ be a connected graph. The notion \emph{the rainbow connection number $rc(G)$} of a graph $G$ was introduced recently by Chartrand et al. Basavaraju et al. showed that for every bridgeless graph $G$ with radius $r$, $rc(G)\leq r(r+2)$, and the bound is tight. In this paper, we prove that if $G$ is a connected graph, and $D^{k}$ is a connected $k$-step dominating set of $G$, then $G$ has a connected $(k-1)$-step dominating set $D^{k-1}\supset D^{k}$ such that $rc(G[D^{k-1}])\leq rc(G[D^{k}])+\max\{2k+1,b_k\}$, where $b_k$ is the number of bridges in $E(D^{k}, N(D^{k}))$. Furthermore, for a connected graph $G$ with radius $r$, let $u$ be the center of $G$, and $D^{r}=\{u\}$. Then $G$ has $r-1$ connected dominating sets $D^{r-1}, D^{r-2},..., D^{1}$ satisfying $D^{r}\subset D^{r-1}\subset D^{r-2} ...\subset D^{1}\subset D^{0}=V(G)$, and $rc(G)\leq \sum_{i=1}^{r}\max\{2i+1,b_i\}$, where $b_i$ is the number of bridges in $E(D^{i}, N(D^{i})), 1\leq i \leq r$. From the result, we can get that if for all $1\leq i\leq r, b_i\leq 2i+1$, then $rc(G)\leq \sum_{i=1}^{r}(2i+1)= r(r+2)$; if for all $1\leq i\leq r, b_i> 2i+1$, then $rc(G)= \sum_{i=1}^{r}b_i$, the number of bridges of $G$. This generalizes the result of Basavaraju et al.