Star-Forest Decompositions of Complete Graphs

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 \(\lceil \frac{n}{2}\rceil +1\) plane star-forests. Additionally we prove that for even n, every decomposition of a 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.