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arxiv: 2301.12783 · v5 · submitted 2023-01-30 · 💻 cs.DS

The Leafed Induced Subtree in chordal and bounded treewidth graphs

Julien Baste This is my paper

Pith reviewed 2026-05-24 09:36 UTC · model grok-4.3

classification 💻 cs.DS
keywords Fully Leafed Induced SubtreeFPT algorithmtreewidthchordal graphsinduced subtreeleaves
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The pith

The Fully Leafed Induced Subtree problem admits an FPT algorithm by treewidth and a polynomial algorithm on chordal graphs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses the Fully Leafed Induced Subtree problem, which asks for an induced subtree with a given number of vertices and at least a given number of leaves. It provides a fixed-parameter tractable algorithm when the parameter is the treewidth of the graph. It also provides a polynomial-time algorithm restricted to chordal graphs. These results build on and generalize polynomial algorithms previously known only for trees and series-parallel graphs. A reader would care because the problem is NP-complete in general, making these restricted cases important for efficient computation.

Core claim

We provide an FPT algorithm parameterized by treewidth. We also provide a polynomial algorithm when the input graph is restricted to be a chordal graph.

What carries the argument

FPT dynamic programming on tree decompositions for bounded treewidth and polynomial-time methods based on chordal graph structure.

If this is right

  • The problem can be solved in f(treewidth) * n^O(1) time.
  • Chordal graphs admit a polynomial-time solution despite the general NP-completeness.
  • Tractability extends from trees and series-parallel graphs to these wider classes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same structural approach may yield tractability results for other hereditary graph classes with bounded clique number or similar decompositions.
  • Implementation on real-world networks with low treewidth could test practical utility for leaf-heavy subtree extraction.

Load-bearing premise

The structural properties of bounded treewidth graphs and chordal graphs allow for the design of efficient algorithms for the Fully Leafed Induced Subtree problem, as generalized from trees and series-parallel graphs.

What would settle it

A concrete bounded-treewidth graph instance where no correct FPT algorithm outputs the right induced subtree with a vertices and at least b leaves, or a chordal graph where any claimed polynomial algorithm exceeds polynomial runtime on some input.

read the original abstract

In the Fully Leafed Induced Subtrees, one is given a graph $G$ and two integers $a$ and $b$ and the question is to find an induced subtree of $G$ with $a$ vertices and at least $b$ leaves. This problem is known to be NP-complete even when the input graph is $4$-regular. Polynomial algorithms are known when the input graph is restricted to be a tree or series-parallel. In this paper we generalize these results by providing an FPT algorithm parameterized by treewidth. We also provide a polynomial algorithm when the input graph is restricted to be a chordal graph.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript claims an FPT algorithm (parameterized by treewidth) and a polynomial-time algorithm (on chordal graphs) for the Fully Leafed Induced Subtree problem: given G, a, and b, decide whether G contains an induced subtree on exactly a vertices with at least b leaves. The results generalize known polynomial-time algorithms on trees and series-parallel graphs; the abstract states that both new algorithms rely on the structural properties of the respective graph classes.

Significance. If correct, the results usefully extend the boundary of tractability for this induced-subtree problem to two well-studied graph classes. The FPT claim leverages standard dynamic programming on tree decompositions, while the chordal claim presumably exploits perfect elimination orders; both are natural and expected routes. Explicit algorithmic constructions on these classes constitute a concrete contribution to the literature on induced-subgraph problems with leaf-count constraints.

minor comments (2)
  1. [Abstract] The abstract states the existence of the algorithms but does not indicate their running times (e.g., the precise f(tw) dependence or the degree of the polynomial for chordal graphs). Adding these bounds would strengthen the summary.
  2. [Introduction] Notation for the target subtree (number of vertices a, number of leaves b) is introduced without an explicit definition of “leaf” in the induced subtree; a short clarifying sentence in §1 would remove any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The report recognizes that the FPT algorithm parameterized by treewidth and the polynomial-time algorithm on chordal graphs usefully extend prior results on trees and series-parallel graphs via standard structural approaches. No specific major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; algorithms rely on standard DP techniques for the graph classes

full rationale

The paper presents FPT and polynomial-time algorithms for the Fully Leafed Induced Subtree problem on bounded-treewidth and chordal graphs, respectively. These rest on well-established structural properties (tree decompositions and perfect elimination orders) that are external to the paper and routinely applied to induced-subgraph problems. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the stated claims or abstract. The derivation chain is self-contained against external algorithmic benchmarks and does not reduce any result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard definitions from graph theory and algorithmic complexity; no new entities or free parameters introduced in the abstract.

axioms (2)
  • standard math The Fully Leafed Induced Subtree problem is NP-complete on 4-regular graphs
    Stated as known in the abstract, relying on prior results.
  • standard math Polynomial algorithms exist for trees and series-parallel graphs
    Cited as known results being generalized.

pith-pipeline@v0.9.0 · 5624 in / 1148 out tokens · 30460 ms · 2026-05-24T09:36:15.637857+00:00 · methodology

discussion (0)

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Reference graph

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