  We deal with the problem of decomposing a complete geometric graph into plane
star-forests. In particular, we disprove a recent conjecture by Pach, Saghafian
and Schnider by constructing for each $n$ a complete geometric graph on $n$
vertices which can be decomposed into $\frac{n}{2}+1$ plane star-forests.
Additionally we prove that for even $n$, every decomposition of complete
abstract graph on $n$ vertices into $\frac{n}{2}+1$ star-forests is composed of
a perfect matching and $\frac{n}{2}$ star-forests with two edge-balanced
components, which we call broken double stars.
