Hardness and Tractability of T_{h+1}-Free Edge Deletion

TL;DR

This paper investigates the computational complexity of the T(h+1)-Free Edge Deletion problem, revealing its hardness under many parameters and identifying specific cases where it remains tractable, thus advancing understanding of its algorithmic boundaries.

Contribution

It proves W[1]-hardness for various structural parameters and identifies new fixed-parameter tractable cases, extending the landscape of known complexity results for the problem.

Findings

W[1]-hardness when parameterized by vertex deletion distance to disjoint unions of paths or stars

Fixed-parameter tractability when parameterized by cluster vertex deletion with h and neighborhood diversity

Existence of FPT algorithms on split and interval graphs

Abstract

We study the parameterized complexity of the T(h+1)-Free Edge Deletion problem. Given a graph G and integers k and h, the task is to delete at most k edges so that every connected component of the resulting graph has size at most h. The problem is NP-complete for every fixed h at least 3, while it is solvable in polynomial time for h at most 2. Recent work showed strong hardness barriers: the problem is W[1]-hard when parameterized by the solution size together with the size of a feedback edge set, ruling out fixed-parameter tractability for many classical structural parameters. We significantly strengthen these negative results by proving W[1]-hardness when parameterized by the vertex deletion distance to a disjoint union of paths, the vertex deletion distance to a disjoint union of stars, or the twin cover number. These results unify and extend known hardness results for treewidth, pathwidth, and feedback vertex set, and show that several restrictive parameters, including treedepth, cluster vertex deletion number, and modular width, do not yield fixed-parameter tractability when h is unbounded. On the positive side, we identify parameterizations that restore tractability. We show that the problem is fixed-parameter tractable when parameterized by cluster vertex deletion together with h, vertex deletion set into a clique and also when parameterized by neighborhood diversity together with h via an integer linear programming formulation. We further present a fixed-parameter tractable bicriteria approximation algorithm parameterized by k. Finally, we show that the problem admits fixed-parameter tractable algorithms on split graphs and interval graphs, and we establish hardness for a directed generalization even on directed acyclic graphs.