Edge deletion to tree-like graph classes

For a fixed property (graph class) H, given a graph G and an integer k, the H -deletion problem consists in deciding if we can turn G into a graph with the property H by deleting at most k edges. The H -deletion problem is known to be NP -hard for most of the well -studied graph classes, such as chordal, interval, bipartite, planar, comparability and permutation graphs, among others; even deletion to cacti is known to be NP -hard for general graphs. However, there is a notable exception: the deletion problem to trees is polynomial. Motivated by this fact, we study the deletion problem for some classes similar to trees, addressing in this way a knowledge gap in the literature. We prove that deletion to cacti is hard even when the input is a bipartite graph. On the positive side, we show that the problem becomes tractable when the input is chordal, and for the special case of quasi -threshold graphs we give a simpler and faster algorithm. In addition, we present sufficient structural conditions on the graph class H that imply the NP -hardness of the H -deletion problem, and show that deletion from general graphs to some well-known subclasses of forests is NP -hard. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).