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 \scrP it is NP-hard to compute the smallest number of edge deletions needed to make a graph satisfy \scrP. Despite being extensively studied in the past 40 years, this problem is still wide open. In fact, it is open even when \scrP is the property of being H-free, for some fixed graph H. In this case we use Rem_H(G) to denote the smallest number of edge deletions needed to turn G into an H-free graph. Alon, Shapira, and Sudakov proved that if H is not bipartite, then computing Rem_H(G) is NP-hard. They left open the problem of classifying the bipartite graphs H for which computing Rem_H(G) is NP-hard. In this paper we resolve this problem when H is a forest, showing that computing 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 Erdős and Sós, which are of independent interest.