REVIEW 3 major objections 3 minor 1 cited by
Edge open packing: further characterizations
T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read For every integer $t\geq 3$, the paper gives necessary and sufficient conditions for a graph to have edge open packing number exactly $t$, and it characterizes the graphs with $\rho_{e}^o(G)=m-3$.
desk verdict A clear, honest abstract claiming to complete a narrow classification; worth refereeing, but unverifiable without the proofs. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the common-edge relation: two edges $e_1,e_2$ have a common edge $e$ when $e$ joins an endpoint of $e_1$ to an endpoint of $e_2$ and is distinct from both. A set of edges in which no two share a common edge is an edge open packing, and $\rho_{e}^o(G)$ is the maximum size of such a set. This relation can be read as open packing in the line graph, and the argument carries the classification by deciding, graph subfamily by subfamily, whether a packing of size $t$ exists and whether an extremal graph can admit $m-3$ edges while excluding $m-2$.
What would settle it
A brute-force enumeration of all simple graphs with up to eight vertices, computing $\rho_{e}^o(G)$ by checking every subset of edges and then testing the paper's two structural conditions, would settle the iff claims: a single graph that has the stated number but fails the conditions, or satisfies the conditions but has a different number, is a counterexample.
Extended reading notes
Core claim
On the paper's own terms, the central claim is a pair of classification theorems. For any integer $t\geq 3$, a graph $G$ satisfies $\rho_{e}^o(G)=t$ exactly when $G$ meets a stated set of structural conditions, and these conditions are necessary as well as sufficient. In the extremal case, the graphs with $\rho_{e}^o(G)=m-3$ are identified by an explicit structural description, extending the known ladder $m,m-1,m-2$ one step further. The abstract states the results but does not sketch the proofs; the theorems are offered as the completion of the classification begun for $t=1,2$.
Load-bearing premise
The load-bearing premise is that the earlier classification of the values $1,2,m,m-1,m-2$ is fully correct under the same definition of 'common edge', and that the proof's exhaustive case analysis over graph subfamilies misses no case; a mistake in either place would break the new necessary-and-sufficient claims.
Editorial extensions
If this is right
- For every finite simple graph, the value of the edge open packing number can be certified by checking the paper's structural conditions, so no exhaustive search over edge subsets is needed to prove that a larger packing is impossible.
- Together with the earlier cases $t=1,2$ and the near-maximum values, the new theorems cover every possible value of the edge open packing number, leaving no gap in the classification.
- The $m-3$ characterization adds a rung to the extremal ladder, so the extremal side of the range is now understood down to $m-3$ rather than stopping at $m-2$.
- Any graph family whose members satisfy or fail the stated conditions has its edge open packing number determined immediately, which gives a direct route to computing the number in structured classes without search.
Reading between the lines
- A natural testable extension is to enumerate all connected simple graphs on up to eight vertices by computer, compute the packing number by brute force, and compare it with the paper's conditions; any mismatch would pinpoint exactly where an iff claim fails.
- Because the edge open packing condition is open packing in the line graph, the paper's structural characterizations likely transfer to a description of line graphs with open packing number $t$, a consequence the abstract does not state.
- The same extremal-ladder method may continue to $m-4$ and $m-5$, but the number of graph subfamilies to check grows; whether a clean pattern persists beyond $m-3$ is left open by this paper.
- If checking the stated conditions is efficient, the paper supplies polynomial-time recognition of graphs with any prescribed edge open packing number; if not, it still provides a finite certificate for each value.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims two results on the edge open packing number rho_e^o(G). First, it gives necessary and sufficient conditions for rho_e^o(G)=t for every integer t>=3. Second, it characterizes graphs satisfying rho_e^o(G)=m-3, where m is the number of edges. The abstract states that these results extend the prior classification by Chelladurai et al. (2022), which covered rho_e^o(G)=1,2 and rho_e^o(G) in {m-2,m-1,m}. The abstract defines the relevant terms and states the theorems, but it contains no proof sketch, lemma statements, or indication of the proof technique.
Significance. If the results are correct, they complete the classification of all possible values of the edge open packing number: the values t=1,2 are covered by earlier work, and Theorem 1 covers every t>=3, while Theorem 2 addresses the high-value case m-3. This would be a useful contribution to structural graph theory. The paper builds directly on a published characterization and appears to introduce no ad-hoc parameters or computational shortcuts, which is a strength. However, because the central claims are exact if-and-only-if characterizations over all finite simple graphs, their correctness depends on exhaustive case analyses, and the significance cannot be fully assessed without the complete proofs.
major comments (3)
- [Abstract (Theorem 1)] The theorem states necessary and sufficient conditions for rho_e^o(G)=t for every integer t>=3, but the abstract contains no proof sketch, no lemma statements, and no description of the proof method. Because this is a classification over all finite simple graphs, the completeness of the case analysis is the load-bearing point, and it cannot be audited from the abstract alone. The full proof is required for a substantive review.
- [Abstract (Theorems 1 and 2)] The relation between the two main results is ambiguous. Once Theorem 1 gives necessary and sufficient conditions for every t>=3, the characterization of rho_e^o(G)=m-3 in Theorem 2 appears to be a special case for t=m-3 (for m>=6). The abstract should state explicitly what additional content Theorem 2 provides, such as a concrete structural description or a treatment of exceptional small cases; otherwise the second theorem risks being presented as a separate contribution when it is a corollary of the first.
- [Abstract (scope of the characterization)] The abstract does not specify the graph class under consideration beyond 'a graph G=(V,E)'. It should state explicitly that all graphs are finite and simple, as is standard for this invariant. In addition, the statement 'characterize the graphs with rho_e^o(G)=m-3' should clarify the domain of m, since for m<3 the value m-3 is negative or zero and cannot equal the edge open packing number of a graph with edges.
minor comments (3)
- [Abstract (definition)] The definition of 'common edge' says that e joins an endpoint of e1 to an endpoint of e2, but it does not explicitly state whether e1 and e2 may be adjacent or share an endpoint; the authors should state that any edge of G satisfying the incidence condition qualifies, including edges adjacent to e1 or e2.
- [Abstract (Theorem 2)] The phrase 'we further characterize the graphs G' is vague; the abstract would benefit from an explicit list of the values of rho_e^o already characterized in Chelladurai et al. (2022) and the values newly characterized here.
- [Abstract (notation)] The symbol rho_e^o(G) is introduced with the phrase 'represented by', which is an unusual way to define notation; the authors should state 'denoted by' or 'written as' for clarity.
Circularity Check
No circularity found: abstract-only review shows a standard extension of a cited prior characterization with no fitted parameters or self-referential definitions.
full rationale
The review covers only the abstract. The paper defines edge open packing sets and the invariant rho_e^o(G) from first principles, then states new necessary and sufficient conditions for rho_e^o(G)=t (t>=3) and a characterization of graphs with rho_e^o(G)=m-3, building on the Chelladurai et al. (2022) results for t=1,2 and m-2,m-1,m. No parameter is fitted, no prediction is made from data, and no construction is defined in terms of the target result. The cited 2022 characterization is prior published work used as a stepping stone, which is standard mathematical practice rather than circular reasoning. Even under the reviewing rule that all manuscript text is in scope, the available abstract provides no specific reduction of a claimed derivation to its own inputs. The central if-and-only-if claims rest on proofs not available for review, but unavailability of proofs is an evidential limitation, not circularity. Therefore the appropriate finding is no significant circularity, score 0.
Assumptions & free parameters
assumptions (2)
- domain assumption The Chelladurai et al. (2022) characterization of edge open packing numbers 1, 2, m, m - 1, and m - 2 is correct under the same definition of common edge.
- standard math Exhaustive case analysis over graph subfamilies is valid and complete in the proofs.
Cite this review
Pith. "Pith review of Edge open packing: further characterizations." pith.science (2026). https://pith.science/paper/32CXQVWG
@misc{pith2026250801935,
author = {Pith},
title = {Pith review of: Edge open packing: further characterizations},
year = {2026},
howpublished = {\url{https://pith.science/paper/32CXQVWG}},
note = {Machine review of arXiv:2508.01935}
}
abstract
Let $G=(V, E)$ be a graph where $V(G)$ and $E(G)$ are the vertex and edge sets, respectively. In a graph $G$, two edges $e_1, e_2\in E(G)$ are said to have \emph{common edge} $e\neq e_1, e_2$ if $e$ joins an endpoint of $e_1$ to an endpoint of $e_2$ in $G$. A subset $D\subseteq E(G)$ is called an \emph{edge open packing set} in $G$ if no two edges in $D$ share a common edge in $G$, and the largest size of such a set in $G$ is known as \emph{edge open packing number}, represented by $\rho_{e}^o(G)$. In the introductory paper (Chelladurai et al. (2022)), necessary and sufficient conditions for $\rho_{e}^o(G)=1, 2$ were provided, and the graphs $G$ with $\rho_{e}^o(G)\in \{m-2, m-1, m\}$ were characterized, where $m$ is the number of edges of $G$. In this paper, we further characterize the graphs $G$. First, we show necessary and sufficient conditions for $\rho_{e}^o(G)=t$, for any integer $t\geq 3$. Finally, we characterize the graphs with $\rho_{e}^o(G)=m-3$.
Forward citations
Cited by 1 Pith paper
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Edge open packing on subclasses of chordal graphs
Claims polynomial-time algorithms for maximum edge open packing on proper interval, block, and split graphs, but the algorithms as written are incomplete or not polynomial.
Reviewed August 6, 2026 · model on record in the stance chip above.
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