Trimming forests is hard (unless they are made of stars)

Graph modification problems ask for the minimal number of vertex/edge additions/deletions needed to make a graph satisfy some predetermined property. A (meta) problem of this type, which was raised by Yannakakis in 1981, asks to determine for which properties ${\mathcal P}$, it is NP-hard to compute the smallest number of edge deletions needed to make a graph satisfy ${\mathcal P}$. Despite being extensively studied in the past 40 years, this problem is still wide open. In fact, it is open even when ${\mathcal P}$ is the property of being $H$-free, for some fixed graph $H$. In this case we use $\text{rem}_{H}(G)$ to denote the smallest number of edge deletions needed to turn $G$ into an $H$-free graph. Alon, Sudakov and Shapira [Annals of Math. 2009] proved that if $H$ is not bipartite, then computing $\text{rem}_{H}(G)$ is NP-hard. They left open the problem of classifying the bipartite graphs $H$ for which computing $\text{rem}_{H}(G)$ is NP-hard. In this paper we resolve this problem when $H$ is a forest, showing that computing $\text{rem}_{H}(G)$ is polynomial-time solvable if $H$ is a star forest and NP-hard otherwise. Our main innovation in this work lies in introducing a new graph theoretic approach for Yannakakis's problem, which differs significantly from all prior works on this subject. In particular, we prove new results concerning an old and famous conjecture of Erd\H{o}s and S\'os, which are of independent interest.