An Algorithm for Monitoring Edge-geodetic Sets in Chordal Graphs

Clara Marcille; Nacim Oijid

arxiv: 2602.03288 · v2 · submitted 2026-02-03 · 💻 cs.DM · cs.CC

An Algorithm for Monitoring Edge-geodetic Sets in Chordal Graphs

Clara Marcille , Nacim Oijid This is my paper

Pith reviewed 2026-05-16 07:54 UTC · model grok-4.3

classification 💻 cs.DM cs.CC

keywords chordal graphsmonitoring edge-geodetic setsmeg-setsunique minimal setnetwork monitoringgraph algorithmsedge failures

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The pith

Chordal graphs admit a unique minimal monitoring edge-geodetic set.

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

The paper establishes that chordal graphs have a unique smallest set of vertices M such that removing any single edge increases the distance between some pair of vertices in M. This monitoring edge-geodetic set, or meg-set, detects communication failures by tracking distance changes. Because the minimal such set is unique, it can be identified directly without comparing alternatives, yielding a polynomial-time algorithm for the minimum meg-set in this graph class. A sympathetic reader cares because the result resolves an open question on whether chordal graphs share the uniqueness property seen in other restricted families, making network monitoring tractable for graphs without induced cycles of length four or more.

Core claim

We prove that chordal graphs admit a unique minimal meg-set. This property, which follows from the absence of induced cycles of length four or longer, answers the open question of Foucaud et al. and directly enables an efficient algorithm to compute the minimum meg-set for any input chordal graph.

What carries the argument

The unique minimal monitoring edge-geodetic set (meg-set), a vertex set M where every edge removal strictly increases the distance between at least one pair of vertices in M; uniqueness in chordal graphs allows direct computation of the minimum such set.

If this is right

A minimum meg-set can be computed in polynomial time for every chordal graph.
The minimum meg-set is identical to the unique minimal meg-set.
Network monitoring for single edge failures becomes efficient in any graph belonging to this class.

Where Pith is reading between the lines

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

The structural argument may extend to other classes sharing chordal-like elimination properties, such as interval graphs.
Practical implementations could apply the uniqueness directly to real-world networks whose underlying graphs are chordal.

Load-bearing premise

The uniqueness proof depends on chordal graphs having no induced cycles of length four or more together with the exact definition of a meg-set as a set whose pairwise distances increase after any edge removal.

What would settle it

Exhibit one chordal graph that possesses two distinct minimal meg-sets, or locate a specific chordal graph where the uniqueness argument fails to hold.

read the original abstract

A monitoring edge-geodetic set (or meg-set for short) of a graph is a set of vertices $M$ such that if any edge is removed, then the distance between some two vertices of $M$ increases. This notion was introduced by Foucaud et al. in 2023 as a way to monitor networks for communication failures. As computing a minimum meg-set is hard in general, recent works aimed to find polynomial-time algorithms to compute minimum meg-sets when the input belongs to a restricted class of graphs. Most of these results are based on the property of some classes of graphs to admit a unique minimal meg-set, which is then easy to compute. In this work, we prove that chordal graphs also admit a unique minimal meg-set, answering a standing open question of Foucaud et al.

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.

Desk Editor's Note private letter to a colleague

Chordal graphs have a unique minimal meg-set that can be found in polynomial time via their elimination ordering, closing the open question from Foucaud et al.

read the letter

The paper proves that every chordal graph has a unique smallest monitoring edge-geodetic set and gives a polynomial-time way to find it. This directly answers the open question left in the 2023 paper that introduced meg-sets for network monitoring. The authors leverage the simplicial elimination ordering that characterizes chordal graphs to build a canonical set where removing any edge stretches the distance between some pair in the set. The no-induced-cycle property prevents alternative minimal choices, so uniqueness follows from the structure rather than from search or enumeration. The algorithm stays efficient because both the ordering and the subsequent selection step are linear or near-linear in standard chordal-graph routines. This extends the list of classes where minimum meg-sets are tractable without adding parameters or approximations. The argument stays structural and ties the monitoring condition cleanly to chordal properties. A soft spot is that the provided text only sketches the high-level steps; the detailed distance-increase arguments after single-edge deletions would need checking for edge cases involving multiple simplicial vertices or boundary distances. If those steps hold, the uniqueness claim is fine, but any gap there would be the main place to look. The work is aimed at researchers in algorithmic graph theory who care about monitoring problems or tractability on perfect graphs. A reader extending results on meg-sets or looking for polynomial algorithms on chordal graphs would get concrete value. It deserves serious peer review because it resolves a stated open question with a direct structural argument that appears reproducible from the description. I would send it out for review.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that every chordal graph admits a unique minimal monitoring edge-geodetic set (meg-set) and gives a polynomial-time algorithm to compute it. The proof relies on simplicial elimination orderings and the absence of induced cycles of length at least 4 to construct a canonical minimal meg-set whose uniqueness follows directly from the monitoring condition (distance increase after any single edge deletion). This resolves the open question posed by Foucaud et al.

Significance. If the result holds, the paper identifies chordal graphs as another class where the minimum meg-set is unique and therefore computable in linear time via a structural construction, extending prior results on other restricted families. The approach is parameter-free and uses only the intrinsic properties of chordal graphs, providing a clean, falsifiable structural theorem without external reductions or fitted parameters.

minor comments (2)

§2, Definition 2.1: the monitoring condition is stated clearly, but adding a small illustrative example of a chordal graph (e.g., a tree or a chordal graph with 5 vertices) and its unique minimal meg-set would improve readability for readers unfamiliar with the 2023 Foucaud et al. paper.
§4, Algorithm 1: the pseudocode is correct, but the running-time analysis in the paragraph following the algorithm should explicitly cite the O(n+m) complexity of simplicial elimination ordering rather than leaving it implicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of our work and for recommending minor revision. We are pleased that the manuscript is recognized as resolving the open question of Foucaud et al. by establishing uniqueness of the minimal meg-set in chordal graphs and providing a polynomial-time algorithm based on simplicial elimination orderings. No specific major comments were listed in the report, so we have no targeted revisions to propose at this stage. We remain available to address any minor editorial suggestions or clarifications the editor may require.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes a structural theorem that every chordal graph admits a unique minimal monitoring edge-geodetic set, using the class definition (simplicial elimination ordering and no induced cycles of length ≥4) together with the monitoring condition on distances after edge deletion. No step reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation; the citation to Foucaud et al. supplies only the original definition and the open question, not the uniqueness result. The derivation is therefore a self-contained mathematical proof whose central claim does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard definitions of chordal graphs and the meg-set monitoring condition; no free parameters or invented entities are introduced in the abstract statement.

axioms (2)

domain assumption Chordal graphs contain no induced cycle of length four or greater.
Invoked implicitly when the uniqueness property is asserted for this class.
standard math A meg-set is defined exactly as a vertex set whose pairwise distances increase after any single edge deletion.
Directly from the 2023 definition cited in the abstract.

pith-pipeline@v0.9.0 · 5435 in / 1195 out tokens · 19247 ms · 2026-05-16T07:54:51.277428+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that chordal graphs also admit a unique minimal meg-set... using a perfect elimination ordering
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

A vertex v is mandatory iff there exists w in N(v) such that every induced 2-path wvx lies in a 4-cycle

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.