  Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey
number} to be the minimum number of vertices $n$ such that any $k$-edge
coloring of $K_n$ contains either a rainbow (all different colored) triangle or
a monochromatic copy of $G$. In this paper, we obtain general upper and lower
bounds on the Gallai-Ramsey numbers for the graph $G = S_t^{r}$ obtained from a
star of order $t$ by adding $r$ extra independent edges between leaves of the
star so there are $r$ triangles and $t - 2r - 1$ pendent edges in $S_t^{r}$. We
also prove some sharp results when $t = 2$.
