Tree-edges deletion problems with bounded diameter obstruction sets

We study the following problem: given a tree G   and a finite set of trees HH, find a subset O of the edges of G   such that G-OG-O does not contain a subtree isomorphic to a tree from HH, and O   has minimum cardinality. We give sharp boundaries on the tractability of this problem: the problem is polynomial when all the trees in HH have diameter at most 5, while it is NP-hard when all the trees in HH have diameter at most 6. We also show that the problem is polynomial when every tree in HH has at most one vertex with degree more than 2, while it is NP-hard when the trees in HH can have two such vertices.The polynomial-time algorithms use a variation of a known technique for solving graph problems. While the standard technique is based on defining an equivalence relation on graphs, we define a quasiorder. This new variation might be useful for giving more efficient algorithm for other graph problems.