  A tree containing exactly two non-pendant vertices is called a double-star.
Let $k_1$ and $k_2$ be two positive integers. The double-star with degree
sequence $(k_1+1, k_2+1, 1, \ldots, 1)$ is denoted by $S_{k_1, k_2}$. If $G$ is
a cubic graph and has an $S$-decomposition, for a double-star $S$, then $S$ is
isomorphic to $S_{1,1}$, $S_{1,2}$ or $S_{2,2}$. It is known that a cubic graph
has an $S_{1,1}$-decomposition if and only if it contains a perfect matching.
In this paper we study the $S_{1,2}$-decomposition of cubic graphs. First, we
present some necessary conditions for the existence of an
$S_{1,2}$-decomposition in cubic graphs. Then we prove that every $\{C_3, C_5,
C_7\}$-free cubic graph of order $n$ with $\alpha(G)= \frac{3n}{8}$ has an
$S_{1,2}$-decomposition, where $\alpha(G)$ denotes the independence number of
$G$. Finally, we obtain some results on the $S_{1,r-1}$-decomposition of
$r$-regular graphs.
