Treewidth-d Deletion Problem

• Treewidth-d-Deletion is defined as finding a minimum vertex set whose removal reduces the graph's treewidth to at most d.
• It generalizes classical problems such as Vertex Cover (d=0) and Feedback Vertex Set (d=1) and employs advanced reduction and kernelization techniques.
• Algorithmic strategies involve dynamic programming, protrusion replacement, and graph separation methods yielding O(log d)-approximation and exact FPT algorithms.

The Treewidth-d-Deletion problem is a fundamental vertex deletion problem in structural graph theory and parameterized algorithmics. Given an undirected graph $G$ and an integer $d \geq 0$, the goal is to find a vertex set $S \subseteq V(G)$ of minimum size such that the treewidth of the induced subgraph $G - S$ is at most $d$. This problem generalizes classical graph problems such as Vertex Cover (the $d=0$ case) and Feedback Vertex Set ($d=1$), and is core to the study of structural graph decompositions, kernelization, and algorithmic graph minors.

1. Formal Problem Statement

Given a graph $G = (V, E)$ and a target treewidth $d \geq 0$, the Treewidth-d-Deletion problem (also known as TW-d-Del or $k$-Treewidth Vertex Deletion) asks for a vertex set $d \geq 0$0 minimizing $d \geq 0$1 such that $d \geq 0$2, where $d \geq 0$3 denotes the treewidth. The parameterized version typically considers $d \geq 0$4 as the parameter, with $d \geq 0$5 either fixed or part of the input.

Formally,

$d \geq 0$6

Equivalently, via a forbidden minor characterization, Treewidth-d-Deletion is an $d \geq 0$7-Deletion problem for $d \geq 0$8, since a graph has treewidth at most $d \geq 0$9 if and only if it excludes the complete graph $S \subseteq V(G)$0 as a minor (Baste et al., 2019).

A weighted version accepts a node-weighted graph and seeks a set $S \subseteq V(G)$1 of minimum total weight such that $S \subseteq V(G)$2 (Włodarczyk, 2024).

2. Approximation and Meta-Algorithmic Frameworks

A central approach to Treewidth-d-Deletion is via an intricate reduction to the $S \subseteq V(G)$3-Subset Vertex Separator problem, leveraging structural graph separation results. The $S \subseteq V(G)$4-Subset Vertex Separator asks, given a graph $S \subseteq V(G)$5, a terminal subset $S \subseteq V(G)$6, and integer $S \subseteq V(G)$7, for a minimum-size set $S \subseteq V(G)$8 such that every connected component of $S \subseteq V(G)$9 contains at most $G - S$0 terminals. Bicriteria approximation algorithms—returning a solution $G - S$1 of size $G - S$2 while allowing each component at most $G - S$3 terminals—are used as subroutines (Gupta et al., 2018).

This reduction, combined with exact FPT algorithms on bounded-treewidth graphs, leads to an $G - S$4-approximation for Treewidth-d-Deletion in polynomial time, with additional efficiency on graph classes excluding a fixed minor (admitting EPTAS under bidimensionality theory).

Sketch of the reduction process (Gupta et al., 2018):

• Maintain a candidate deletion set $G - S$5.
• Use structure theorems for small separators in bounded-treewidth graphs to relate separators to the $G - S$6-Subset Vertex Separator instance.
• Iteratively refine $G - S$7 using bicriteria separator solutions and exact processing on bounded-treewidth components until a global $G - S$8-size solution is obtained.

For weighted instances, a combinatorial constant-factor approximation is achieved using random protrusion sampling, modulator-hitting families, and martingale analysis. Every fixed $G - S$9 admits a randomized constant-factor approximation in time $d$0 (Włodarczyk, 2024).

3. Kernelization and Compression

Early kernelization results established that, when $d$1 contains a planar graph, the $d$2-Deletion problem admits a polynomial kernel whose size bound is $d$3 for computable functions $d$4 [FOCS 2012]. In the case of Treewidth-d-Deletion, this yields a polynomial kernel with an exponent dependent on $d$5, and a lower bound shows this dependence is necessary unless $d$6 (Sharma et al., 13 Jan 2026).

Recent advancements provide uniform lossy kernels. For every fixed $d$7:

• There exists a 2-approximate kernel of size $d$8, i.e., a polynomial-size reduction to $d$9 vertices incurring at most a factor-2 in the approximation (Sharma et al., 13 Jan 2026).
• A $d=0$0-approximate compression protocol is possible, using $d=0$1 calls to oracles for Treewidth-d-Deletion and $d=0$2-Deletion on graph instances of size polynomial in $d=0$3; the approximation factor can be made arbitrarily close to 1, at the cost of increased compressed size.

A linear kernel is achievable on graph classes excluding a fixed topological minor, for fixed $d=0$4, thus generalizing to sparse graph classes with even disconnected forbidden minors (Sharma et al., 13 Jan 2026). This relies on global protrusion decompositions and extended replacement techniques.

4. Fixed-Parameter and Exact Algorithms

For general graphs, the Treewidth-d-Deletion problem is connected to $d=0$5-M-Deletion with $d=0$6: $d=0$7 has treewidth $d=0$8 if and only if it excludes $d=0$9 as a minor (Baste et al., 2019). The major algorithmic advances follow the structure below:

• Dynamic programming over tree decompositions using the theory of boundaried graphs and equivalence classes for gluing operations (using representatives $d=1$0).
• The Flat Wall Theorem and bidimensionality to obtain small constant-treewidth “modulators” and identify large-wall obstacles.
• The irrelevant vertex technique inside large grid-like subgraphs, enabling reduction to minimal representatives.
• Protrusion replacement: converting protrusions to smaller equivalents, preserving the solution structure.

An optimal algorithm for this family of problems runs in time $d=1$1, where $d=1$2 is the width of the tree decomposition of $d=1$3. The lower bound under ETH rules out a $d=1$4 algorithm even for small $d=1$5 (Baste et al., 2019).

5. Special Cases, Generalizations, and Hardness

Several special cases arise as $d=1$6 varies:

• For $d=1$7 (Vertex Cover), NP-hardness of approximation within $d=1$8 holds [Dinur–Safra], even for exact or parameterized kernels.
• For $d=1$9 (Feedback Vertex Set), the problem remains W[1]-hard under various generalizations.

The generalized framework extends directly to problems such as Pathwidth-, Treedepth-, and minor/transversal versions under hereditary graph classes with bounded treewidth, often with similar algorithmic and kernelization results (Gupta et al., 2018).

Hardness results include:

• For hereditary classes, the maximization version (max $G = (V, E)$0 s.t. tw$G = (V, E)$1) is $G = (V, E)$2-hard for some $G = (V, E)$3.
• No $G = (V, E)$4-approximation independent of $G = (V, E)$5 is known unless a constant-factor $G = (V, E)$6-Subset Vertex Separator algorithm is found.

6. Structural and Technical Ingredients

Key algorithmic and structural tools include:

• Near-protrusion and true protrusion decompositions to partition the graph into bounded connectivity/complexity parts.
• The large-simplicial-neighborhood rule for effective reduction, yielding lossy kernels (Sharma et al., 13 Jan 2026).
• Augmented graphs to facilitate distinction between simplicial and non-simplicial components.
• Exhaustive families of partial solutions for protrusion handling, relying on the theory of monadic second-order logic on bounded-treewidth graphs (Courcelle's theorem).
• Martingale analysis in weighted deletion for bounding expected cost against the optimum (Włodarczyk, 2024).

7. Summary Table of Key Algorithmic Results

Result Type	Bound/Guarantee	Reference
$G = (V, E)$7-approximation	Polynomial time for arbitrary graphs	(Gupta et al., 2018)
Constant-factor apx (weighted)	Randomized, polynomial in $G = (V, E)$8	(Włodarczyk, 2024)
2-approximate kernel (uniform)	$G = (V, E)$9 vertices, polynomial time	(Sharma et al., 13 Jan 2026)
$d \geq 0$0-lossy protocol	Compression with $d \geq 0$1 small-oracle calls, $d \geq 0$2-approximation	(Sharma et al., 13 Jan 2026)
Linear-size exact kernelization	O$d \geq 0$3 vertices on topological-minor-free classes, fixed $d \geq 0$4	(Sharma et al., 13 Jan 2026)
Exact, bounded-treewidth graphs	$d \geq 0$5	(Baste et al., 2019)

All polynomial kernelizations for the general Treewidth-d-Deletion problem are necessarily non-uniform in $d \geq 0$6 unless $d \geq 0$7 (Sharma et al., 13 Jan 2026), indicating a strong lower bound on compressibility.

• (Gupta et al., 2018) "Losing Treewidth by Separating Subsets"
• (Włodarczyk, 2024) "Losing Treewidth In The Presence Of Weights"
• (Baste et al., 2019) "Hitting minors on bounded treewidth graphs. IV. An optimal algorithm"
• (Sharma et al., 13 Jan 2026) "Protrusion Decompositions Revisited: Uniform Lossy Kernels for Reducing Treewidth and Linear Kernels for Hitting Disconnected Minors"