On edge-decomposition of cubic graphs into copies of the double-star with four edges

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 $(k_1+1,k_2+1,1,\dots ,1)$ 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 ${C_3, C_5, C_7}$-freecubicgraphoforder$n$with$\alpha(G)= \frac{3n}{8}$hasan$S_{1,2}$-decomposition,where$\alpha(G)$denotestheindependencenumberof$G$\mathrm{.}Finally,weobtainsomeresultsonthe$S_{1,r-1}$-decompositionof$r$-regular graphs.