Star-Forest Decompositions of Complete Graphs

TL;DR

The paper investigates the minimal number of plane star-forests needed to decompose complete geometric graphs, introduces SF-extendable constructions, and builds on prior work to disprove a conjecture by Pach, Saghafian and Schnider by showing that for every $n$ there exists a complete geometric graph on $n$ vertices that decomposes into $\frac{n}{2}+1$ plane star-forests. It also proves a tight structural result for even $n$: any decomposition of the abstract complete graph $K_n$ into $\frac{n}{2}+1$ star-forests must be a broken double stars decomposition, consisting of a perfect matching and $\frac{n}{2}$ edge-balanced two-star forests whose centers lie at the endpoints of a matching edge. The authors develop a constructive framework (SF-extendable line-segment arrangements, staircases, comets) to generate infinite families of such decompositions and explore computational and geometric properties of these configurations. Overall, the work settles the negative resolution of the conjecture, identifies the tightness of the bound, and highlights the rich interplay between geometric arrangements and abstract decompositions in star-forest partitions.

Abstract

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.