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

Ajinkya Gaikwad, Soumen Maity, Leeja R

TL;DR

This work clarifies the parameterized complexity of Th+1-Free Edge Deletion for unbounded h, proving W[1]-hardness under several restrictive structural parameters (including twin cover, vdp, and vds) and unifying prior hardness results with respect to treewidth/pathwidth. On the positive side, it identifies FPT regimes (cluster vertex deletion plus h, and neighborhood diversity plus h) and provides an FPT bicriteria approximation, along with FPT algorithms on split and interval graphs. It also shows a W[2]-hardness barrier for the directed variant on DAGs, broadening the hardness landscape. The results collectively delineate when fixed-parameter tractability can be achieved and where it fails, guiding future study on restricted graph classes and approximation guarantees.

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, 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.

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

TL;DR

This work clarifies the parameterized complexity of Th+1-Free Edge Deletion for unbounded h, proving W[1]-hardness under several restrictive structural parameters (including twin cover, vdp, and vds) and unifying prior hardness results with respect to treewidth/pathwidth. On the positive side, it identifies FPT regimes (cluster vertex deletion plus h, and neighborhood diversity plus h) and provides an FPT bicriteria approximation, along with FPT algorithms on split and interval graphs. It also shows a W[2]-hardness barrier for the directed variant on DAGs, broadening the hardness landscape. The results collectively delineate when fixed-parameter tractability can be achieved and where it fails, guiding future study on restricted graph classes and approximation guarantees.

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, 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.
Paper Structure (11 sections, 17 theorems, 32 equations, 6 figures)

This paper contains 11 sections, 17 theorems, 32 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. Notation and Definitions
  4. W[1]-hardness parameterized by twin cover number
  5. FPT Parameterized by Cluster Vertex Deletion and h
  6. FPT Parameterized by Neighborhood Diversity and h
  7. An FPT Approximation Algorithm Parameterized by Solution Size
  8. Th+1-Free Edge Deletion on Split Graphs
  9. Th+1-Free Edge Deletion on Interval Graphs
  10. Conclusion
  11. W[1]-hardness parameterized by treedepth

Key Result

Theorem 1

Let $\mathcal{F}$ be any graph class that contains a connected graph on $s$ vertices for every $s\in\mathbb{N}$. Then $\mathcal{T}_{h+1}$-Free Edge Deletion is W[1]-hard when parameterized by the vertex deletion distance of the input graph to $\mathcal{F}$.

Figures (6)

  • Figure 1: Relationship between vertex cover [vc] (see Definition \ref{['defvc']}), neighborhood diversity [nd] (see Definition \ref{['defnd']}), twin cover [tc] (see Definition \ref{['deftc']}), modular width [mw] (see defmodwidth), cluster vertex deletion number [cvd] (see Definition \ref{['cvd def']}), feedback vertex set [fvs] (see Definition \ref{['deffvs']}), pathwidth [pw] (see Definition \ref{['defpw']}), treewidth [tw] (see Definition \ref{['deftw']}) and clique width [cw] (see bib14). We additionally include vertex deletion distance to a disjoint union of paths [vdp] and vertex deletion distance to a disjoint union of stars [vds]. Note that $A\rightarrow B$ means that there exists a function $f$ such that for all graphs, $f(A(G))\geq B(G)$. This gives complexity landscape of $\mathcal{T}_{h+1}$-Free Edge Deletion under different parameterizations. Red indicates W[1]-hardness, green indicates fixed-parameter tractability, and black denotes parameterizations for which the complexity status remains open.
  • Figure 2: The graph $G$ constructed from the instance $I$ of Unary Bin Packing in Theorem \ref{['thm:distF']}. A double edge between a vertex $v$ and a set $S$ denotes that $v$ is adjacent to every vertex in $S$.
  • Figure 3: The graph in the proof of Theorem \ref{['theorem-W[2]']} constructed from Hitting Set instance $U=\{x_{1},x_{2},x_{3},x_{4}\}$, $F= \{ \{x_{1},x_{2}\},\{x_{2},x_{3}\},\{x_{3},x_{4}\}\}$ and $k=2$.
  • Figure 4: The graph $G$ constructed from the instance $I$ in the proof of Theorem \ref{['split']}. A double edge between a vertex $v$ and a set $S$ denotes that $v$ is adjacent to every vertex in $S$. A double edge between a set $A$ and a set $B$ denotes that every vertex in $A$ is adjacent to every vertex in $B$.
  • Figure 5: An example input graph $G=(V, E, w)$ for an MMO instance.
  • ...and 1 more figures

Theorems & Definitions (54)

  • Definition 1
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  • Definition 6
  • Definition 7: Robertson and Seymour Neil
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  • Definition 10
  • ...and 44 more