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arxiv: 2604.24325 · v1 · submitted 2026-04-27 · 💻 cs.DS · cs.CC· math.CO

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Identification to Subclasses of Chordal Graphs

Petr A. Golovach , Laure Morelle , Dani\"el Paulusma
Authors on Pith no claims yet

Pith reviewed 2026-05-08 01:38 UTC · model grok-4.3

classification 💻 cs.DS cs.CCmath.CO
keywords chordal graphsvertex identificationgraph modificationparameterized complexityNP-completenessFPT algorithmssimplicial vertices
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The pith

The complexity of turning a graph into a fixed chordal subclass via at most k vertex identifications is settled for most target classes under parameters k and n-k.

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

The paper studies the H-Identification problem: given a graph G and a fixed subclass H of chordal graphs, decide whether G can be turned into a member of H by merging vertices at most k times, where each merge replaces two vertices with one whose neighborhood is their union. The authors classify both the classical complexity and the parameterized complexity with respect to k and with respect to n-k for a broad collection of such H classes. They also classify the two-graph Identification problem when parameterized by the number of simplicial vertices in the target graph. A sympathetic reader cares because the results give a nearly complete map of when these vertex-merging problems are tractable and when they are hard, which directly informs algorithmic approaches to graph simplification and editing.

Core claim

We determine the classical and parameterized complexity of the H-Identification problem for various subclasses H of chordal graphs, obtaining an almost complete picture for the parameters k and n-k. We also determine the parameterized complexity of the Identification problem when H belongs to one of our testbed classes, using the number of simplicial vertices of H as the parameter.

What carries the argument

The vertex identification operation, which replaces two vertices u and v by a single vertex adjacent to every neighbor of u or v, and the decision problem of whether at most k such operations suffice to reach a graph in the fixed target class H.

If this is right

  • For some fixed H the problem is solvable in polynomial time, while for others it is NP-complete.
  • The same dichotomy holds when the problem is parameterized by k: some cases are FPT and others are para-NP-hard.
  • An analogous split occurs when the parameter is n-k instead of k.
  • The two-graph Identification problem is fixed-parameter tractable when parameterized by the number of simplicial vertices in the target graph.

Where Pith is reading between the lines

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

  • The same classification technique might extend to non-chordal target classes such as planar graphs or graphs of bounded treewidth.
  • The results suggest that vertex identification behaves differently from vertex deletion or edge contraction as an editing operation, which could be tested by comparing running times on the same instances.
  • Practical implementations for the tractable cases could be used to simplify large networks while preserving chordal structure.

Load-bearing premise

The target classes H are fixed chordal subclasses for which standard properties such as the behavior of simplicial vertices and the effect of neighborhood unions hold without further hidden constraints.

What would settle it

A concrete counterexample would be a specific chordal subclass H together with an infinite family of input graphs where the minimum number of identifications needed to reach H lies outside the tractability or hardness classification given by the paper.

Figures

Figures reproduced from arXiv: 2604.24325 by Dani\"el Paulusma, Laure Morelle, Petr A. Golovach.

Figure 1
Figure 1. Figure 1: An example of a graph G that can be identified to P4, as shown with two different P4-witness structures. Note that G is disconnected and cannot be contracted to P4. For graphs G and H, a function f : V (G) → V (H) is called a homomorphism from G to H if f(u)f(v) ∈ E(H) for every edge uv ∈ E(G). A homomorphism f : V (G) → V (H) is a compaction if for each edge xy ∈ E(H) with x ̸= y, there exists an edge uv … view at source ↗
Figure 2
Figure 2. Figure 2: The graph G′ constructed in the proof of Theorem 1. Suppose that (G, p) is a yes-instance of Independent Set. Let I be an independent set in G of size p. Then, by identifying all vertices in G′ − I to a single vertex u ′ , we obtain a star on p + 1 vertices. So, (G′ , n − (p + 1)) is a yes-instance of H-Identification when H is the class of stars, trees, or forests. 5 view at source ↗
Figure 3
Figure 3. Figure 3: The graph G constructed in the proof of Theorem 5. Let U be a universe with n elements, S be a family of m subsets of U, and k ∈ N. We construct a graph G on 2nm + 2m − k vertices as follows (cf view at source ↗
Figure 4
Figure 4. Figure 4: Illustration for Claim 14 C and the clique K′ obtained from K by identifying the vertices of Y and deleting the vertices x with W(x) ⊆ V (C), which is a cluster graph. Since for every x ∈ Y , W(x) contains at least one vertex of U, |W′ (Y )| − 1 = | S x∈Y W(x) \U| − 1 ≤ P x∈Y (|W(x)| − 1), and the number of identifications needed to obtain this graph does not exceed the number of identifications for K. How… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the graph G′ and the graph H in the proof of Theorem 9. and W(xb) = {b} or, symmetrically, W(x ′ a ) = {b} and W(xb) = {a ′}. However, in the second case, we have that the vertices of G are in W(xa) because they are adjacent to b. This contradicts the fact that W is an H-witness structure since |Q| = n − k ≥ 2. Thus, W(x ′ a ) = {a ′} and W(xb) = {b}. This implies that a ∈ W(xa). Since a is… view at source ↗
read the original abstract

An identification of two vertices $u$ and $v$ in a graph replaces them with a new vertex whose neighborhood is the union of the neighborhoods of $u$ and $v$. We study the {\sc ${\cal H}$-Identification} problem, which is to decide whether a given graph $G$ can be transformed (``identified'') to a graph in ${\cal H}$ by applying at most $k$ vertex identifications. We determine the classical and parameterized complexity of this problem for various subclasses ${\cal H}$ of chordal graphs, obtaining an almost complete picture for two parameters: $k$ and $n-k$. We also consider the {\sc Identification} problem, which is to test for two given graphs $G$ and $H$ if $G$ can be identified to $H$. We determine the parameterized complexity of this problem when $H$ is a graph from one of our testbed classes, taking the number of simplicial vertices of $H$ as the parameter.

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 studies the H-Identification problem: given a graph G and integer k, decide whether at most k vertex identifications suffice to produce a graph belonging to a fixed subclass H of chordal graphs. It determines the classical and parameterized complexity of this problem for multiple concrete H (including interval, split, and other chordal subclasses), claiming an almost complete classification under both the parameter k and the dual parameter n-k. It further analyzes the two-graph Identification problem (given G and H, decide if G can be identified to H) when H belongs to the testbed classes, parameterized by the number of simplicial vertices of H.

Significance. If the reductions and algorithms are correct, the work supplies a detailed complexity map for a natural graph-modification operation on well-structured classes, extending the literature on editing to chordal graphs and their subclasses. The parameterized result for the non-fixed-H case, using the number of simplicial vertices, introduces a potentially useful parameter that may transfer to other modification problems.

minor comments (2)
  1. [Abstract / Introduction] The abstract and introduction repeatedly refer to an 'almost complete picture' for the complexity classification; the manuscript should explicitly list the remaining open cases (by class and parameter) in a dedicated table or subsection so that readers can immediately see the precise scope of the results.
  2. [Preliminaries] The definition of vertex identification and the preservation of chordality under identification are used throughout; a short self-contained paragraph recalling the precise neighborhood-union operation and citing the relevant chordal-graph properties would improve readability for readers outside the immediate sub-area.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their concise and accurate summary of our work, as well as for the positive assessment of its significance. We are pleased that the referee recommends minor revision and note that no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper classifies the classical and parameterized complexity of H-Identification for fixed chordal subclasses H via standard polynomial-time reductions to known problems (such as Clique Cover or Vertex Cover variants) and FPT algorithms parameterized by k or n-k. No equations, fitted parameters, self-referential definitions, or load-bearing self-citations appear; the derivation chain consists of explicit reductions whose correctness rests on independent graph-theoretic facts about simplicial vertices and chordal graphs rather than on any input that is renamed or fitted to produce the claimed result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definitions of graphs, chordal graphs, simplicial vertices, and the identification operation, together with the usual framework of classical and parameterized complexity; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (2)
  • standard math Standard definitions of undirected graphs, neighborhoods, induced cycles, chordal graphs, and simplicial vertices.
    These are foundational concepts from graph theory used throughout the paper.
  • domain assumption The parameterized complexity framework with parameters k and n-k and the notion of simplicial vertices as a parameter.
    Assumes the standard definitions of FPT, W[1]-hardness, and kernelization from parameterized complexity theory.

pith-pipeline@v0.9.0 · 5476 in / 1430 out tokens · 93362 ms · 2026-05-08T01:38:26.723910+00:00 · methodology

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