Dismountability in Temporal Cliques Revisited

Arnaud Casteigts; Daniele Carnevale; Timoth\'ee Corsini

REVIEW 3 major objections 5 minor 19 references

Dismountability in Temporal Cliques Revisited

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Dismountability alone recovers the O(n log n) spanner bound for temporal cliques and characterizes any 4n counterexample.

desk verdict Clean structural collapse theorem for k-hop dismountability, but Corollary 4.1 overclaims an equality that is false; the bi-clique reduction needs a small repair. read the letter →

arxiv 2502.01321 v1 pith:OG5FBJN7 submitted 2025-02-03 cs.DM cs.DCcs.NI

Daniele Carnevale , Arnaud Casteigts , Timothée Corsini This is my paper · ORCID

classification cs.DMcs.DCcs.NI MSC 05C8268R10

keywords temporalgraphscliquesspannersdismountabilitypivotabilitydynamicnetworksreachabilityfull-rangelabelings

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that one structural principle, dismountability, is enough to explain why every temporal clique has a sparse temporal spanner. It characterizes, in a necessary and sufficient way, the structure of a temporal clique that resists dismounting at 1, 2, or 3 hops, and proves that if a clique is $k$-hop dismountable for any $k>3$, then it is already $\{1,2,3\}$-hop dismountable. From this, the known $O(n \log n)$ spanner bound follows through a single constructive algorithm that uses only dismountability. A by-product is that any minimal counterexample to a $4n$-edge spanner would have to be a clique that is not dismountable at any hop count. The paper also connects dismountability to pivotability and identifies full-range labelings as a family for which both properties hold.

What carries the argument

The central object is the dismountability of a node $u$ in a temporal clique: if some node $v$ has $u$ as its earliest neighbor and some node $w$ has $u$ as its latest neighbor, then $u$ can be removed from the instance at the cost of keeping the two edges $uv$ and $uw$ in the spanner, and one can recurse on the smaller clique; $k$-hop dismountability relaxes 'earliest/latest neighbor' to temporal paths of length at most $k$. The argument is carried by the partition of vertices into $V^-$ (nodes that are earliest neighbors), $V^+$ (nodes that are latest neighbors), and $V^0$ (the rest), by the two perfect matchings $M^-$ and $M^+$ forced between $V^-$ and $V^+$ in non-$\{1,2\}$-hop-dismountable cliques, and by the label-order condition that forbids an intra-part edge whose label lies between the two incident matching-edge labels. This machinery shows exactly where a recursive dismounting gets stuck.

What would settle it

Run an exhaustive search over all simple, proper labelings of temporal cliques on at most six vertices: if any clique is $k$-hop dismountable for some $k>3$ but has no $\{1,2,3\}$-hop dismountable node, Theorem 3.7 is false. An explicit single such clique would also refute the paper's main structural claim.

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Extended reading notes

Core claim

The central claim is Theorem 3.7: for a simple, proper temporal clique, $k$-hop dismountability for any $k>3$ implies $\{1,2,3\}$-hop dismountability. Thus the only dismountability structure that can obstruct recursion is the hierarchy of 1-, 2-, and 3-hop failures. Combined with the bi-clique reduction of Angrick et al., this gives a proof that every temporal clique admits an $O(n \log n)$-edge temporal spanner using dismountability alone, and any minimal counterexample to $4n$-edge spanners must be non-$k$-hop dismountable for every $k$ and satisfy the complete structural description of Theorem 3.10. The paper further shows that recursively $k$-hop dismountable cliques are pivotable, hence admit $2n-3$ spanners, and that full-range temporally connected graphs are pivotable.

Load-bearing premise

The entire analysis assumes the standard reduction to temporal cliques in which every edge has exactly one time label and adjacent edges have different labels, because extra labels or equal adjacent labels are said only to increase reachability; if that reduction fails in the strict-path setting, the characterizations would not cover the general problem.

Editorial extensions

If this is right

Any temporal clique that is $k$-hop dismountable for $k>3$ is already $\{1,2,3\}$-hop dismountable, so no new dismountability structure appears beyond three hops.
The $O(n \log n)$ spanner bound for temporal cliques follows constructively from dismountability alone, via a polynomial-time algorithm.
Every minimal counterexample to $4n$-edge spanners must be non-$k$-hop dismountable for all $k$ and must satisfy the complete structural description of Theorem 3.10.
Non-$\{1,2\}$-hop dismountable cliques reduce to temporal bi-cliques on the same vertex set, preserving the vertex count for later constant-factor studies.
Recursively $k$-hop dismountable cliques admit $2n-3$ edge spanners, and full-range temporally connected graphs admit $O(n)$ spanners via pivotability.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the $4n$ counterexample structure is ever shown impossible, the proof would likely come from analyzing non-$\{1,2,3\}$-hop dismountable cliques only; the paper's characterization narrows the search space to that class.
The paper's finding that full-range (large-lifetime) labelings are easy suggests hard instances are concentrated at small compressed lifetimes, so decomposing a labeling by lifetime could be a route toward an $O(n)$ bound.
The implication that recursive dismountability yields pivotability offers a possible unification: any recursively dismountable clique gets a linear spanner through a single pivot edge.
A testable extension is whether every temporal clique admits a 2-arch spanner (a maximal 2-degenerate footprint), as preliminary experiments in the paper suggest; if true, that would strengthen the evidence for linear-size spanners beyond dismountability.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The real content of this paper is the collapse theorem (Theorem 3.7): if a temporal clique is k-hop dismountable for any k>3, it is {1,2,3}-hop dismountable. Combined with the characterizations of non-{1,2}- and non-{1,2,3}-hop dismountable cliques (Theorems 3.9 and 3.10), this gives a clean picture: only the 1,2,3 hierarchy matters. Corollary 3.8—any minimal counterexample to 4n spanners must be non-k-hop dismountable—is a nice by-product. The recovery of the O(n log n) bound using only dismountability is genuinely elegant, though it leans on the Angrick et al. bi-clique result.

The paper is honest about not resolving the O(n) question, and the structural theorems appear correct after careful reading. The proofs are mostly well-argued.

The soft spots are localized but real. Corollary 4.1 overclaims: it states M− := {e−(v): v∈V+} equals {e−(v): v∈V−}. That is false. A 4-node example suffices: V−={a,b}, V+={x,y}, with labels ab=1, ax=2, by=3, ay=4, bx=5, xy=6. This is simple, proper, and satisfies the Theorem 3.9 conditions, yet {e−(v): v∈V+}={ax,by} while {e−(v): v∈V−}={ab}. The same holds for M+. The reduction only needs the two sets to be perfect matchings between the parts, and the 'prefixing/suffixing' argument can be re-derived from that, but as written the formal statement is wrong and the subsequent paragraph is too terse to cover the gap. Also, Theorem 5.2's induction is sketched rather than fully written out, and Theorem 5.7's 'can be composed with it' needs a sentence more. These are presentation defects, not cracks in the main structural results.

The simple/proper assumption is standard and the paper defends it; I don't read that as a flaw.

Overall: this deserves a serious referee. The collapse theorem and characterizations are worth publishing; they will be useful to anyone working on temporal spanners. The revision should fix Corollary 4.1 and tighten the two proofs mentioned. I'd accept it with major/minor revisions rather than desk-reject.

Referee Report

3 major / 5 minor

Summary. The paper revisits the dismountability technique for temporal cliques. It gives structural characterizations of cliques that are not 1-hop dismountable, not {1,2}-hop dismountable, and not {1,2,3}-hop dismountable, and proves that if a clique is k-hop dismountable for any k > 3 then it is already {1,2,3}-hop dismountable (Theorem 3.7). Based on this, the paper proposes a reduction from cliques to bi-cliques using only dismountability, recovering the O(n log n) temporal spanner bound. In a second part, it proves that recursively k-hop dismountable cliques are pivotable and introduces full-range temporal graphs, showing that full-range cliques are {1,2,3}-hop dismountable and full-range temporally connected graphs are pivotable.

Significance. If the main structural claims and reductions are correct, this is a valuable simplification of the known O(n log n) spanner construction for temporal cliques, and the structural characterization of minimal counterexamples to linear spanners is of independent interest. The paper is proof-oriented and does not rely on numerical fitting or black-box parameter tuning; its claims are derived from the definitions. The pivotability result connecting recursive dismountability to pivotability is a useful addition, and the full-range family is a genuinely new class with clean properties. However, several formal statements in the proof chain need correction before the paper can be considered reliable.

major comments (3)

[Section 4, Corollary 4.1]
[Section 3.4, Theorem 3.7]
[Section 3.5, Theorem 3.10]

minor comments (5)

[Abstract]
[Section 3.4, Theorem 3.7 proof]
[Section 5, Theorem 5.2 proof]
[Section 5.1, Theorem 5.7 proof]
[Section 4.1]

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the structural theorems are proved from the definitions, and the cited prior results are external lemmas rather than fitted inputs or self-referential assumptions.

full rationale

The paper's central derivation chain is self-contained. Theorem 3.7, Corollary 3.8, and the characterizations in Theorems 3.9 and 3.10 are obtained by direct arguments from the definitions of e−(v), e+(v), V−, V+, and k-hop dismountability; the characterizations are proved in both directions instead of being imposed as definitions. The O(n log n) recovery in Section 4 relies on the bi-clique spanner theorem of Angrick et al. [3] and on the original dismountability lemma of Casteigts et al. [13], but these are external, peer-reviewed results with stated assumptions that do not include the paper's collapse theorem, so citing them is legitimate support rather than circularity. There are no fitted parameters, no quantity is fitted to data and then 'predicted', and no uniqueness or structural conclusion is imported from the authors' own prior work as a substitute for proof. The skeptical note about Corollary 4.1 concerns a possible mathematical overclaim in the asserted equality of two sets of earliest edges; even if that equality is false as stated, it is a correctness defect in a formal statement and not a circular step, because the bipartite reduction's usable matching properties do not reduce to the equality being used as its own input. In summary, no step of the derivation is equivalent by construction to its own assumptions.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters. The central claims rest on standard graph theory, the simple/proper temporal clique convention, and black-box prior results ([13] and [3]). The full-range family is a new definition, not an empirical entity.

assumptions (6)

domain assumption Temporal cliques are restricted to simple and proper labelings, which is without loss for the spanner problem (Section 1.1).
All structural lemmas assume single-valued locally injective labels; if multiple labels or equal adjacent labels are allowed, earliest and latest edges may not be unique and dismountability arguments break.
domain assumption Theorem 2.1 (1-hop dismountability) and its k-hop generalization from Casteigts et al. [13] are correct.
The paper builds on these prior results; the spanner-preserving recursion is imported.
domain assumption Angrick et al. [3] proved that temporal bi-cliques admit O(n log n) spanners via one-sided dismountability and a splitting recurrence (Section 4.1).
Used as a black box to complete the O(n log n) proof; the present paper does not re-derive this.
domain assumption Pivot edge implies a 2n-3 spanner, from Bui-Xuan et al. [10] (Section 5).
Used for Corollary 5.3 and pivotability connections.
domain assumption Time compression preserves reachability (Section 5.1).
Full-range graphs are defined on compressed labelings; if compression changed reachability, the pivotability results would not transfer.
standard math Standard induction and the master theorem are used.
Used in Lemma 5.6 and Section 4.1 recurrence.

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Pith. "Pith review of Dismountability in Temporal Cliques Revisited." pith.science (2026). https://pith.science/paper/OG5FBJN7

@misc{pith2026250201321,
  author       = {Pith},
  title        = {Pith review of: Dismountability in Temporal Cliques Revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OG5FBJN7}},
  note         = {Machine review of arXiv:2502.01321}
}

abstract

A temporal graph is a graph whose edges are available only at certain points in time. It is temporally connected if the nodes can reach each other by paths that traverse the edges chronologically (temporal paths). In general, temporal graphs do not always admit small subsets of edges that preserve connectivity (temporal spanners). In the case of temporal cliques, spanners of size $O(n\log n)$ are guaranteed. The original proof by Casteigts et al. [ICALP 2019] combines a number of techniques, one of which is dismountability. In a recent work, Angrick et al. [ESA 2024] simplified the proof and showed, among other things, that a one-sided version of dismountability can be used to replace the second part of the proof. In this paper, we revisit the dismountability principle. We characterizing the structure that a temporal clique has if it is not 1-hop dismountable, then not {1,2}-hop dismountable, and finally not {1,2,3}-hop dismountable. It turns out that if a clique is k-hop dismountable for any other k, then it must also be {1,2,3}-hop dismountable. Interestingly, excluding only 1-hop and 2-hop dismountability is already sufficient for reducing the spanner problem from cliques to bi-cliques. Put together with the strategy of Angrick et al., the entire $O(n \log n)$ result can now be recovered using only dismountability. An interesting by-product of our analysis is that any minimal counter-example to the existence of $4n$ spanners must satisfy the properties of non {1,2,3}-hop dismountable cliques. In the second part, we discuss connections between dismountability and pivotability. We show that recursively k-hop dismountable cliques are pivotable (and thus admits $2n$ spanners, whatever k). We define a family of labelings (called full-range) which force both dismountability and pivotability and that gives some evidence that large lifetimes could be exploited more generally.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Akrida, Leszek Gasieniec, George B

Eleni C. Akrida, Leszek Gasieniec, George B. Mertzios, and Paul G. Spirakis. The complexity of optimal design of temporally connected graphs. Theory of Computing Systems , 61(3):907--944, 2017

work page 2017

[2] [2]

Constructing light spanners deterministically in near-linear time

Stephen Alstrup, S ren Dahlgaard, Arnold Filtser, Morten St \"o ckel, and Christian Wulff-Nilsen. Constructing light spanners deterministically in near-linear time. Theoretical Computer Science , 907:82--112, 2022

work page 2022

[3] [3]

How to reduce temporal cliques to find sparse spanners

Sebastian Angrick, Ben Bals, Tobias Friedrich, Hans Gawendowicz, Niko Hastrich, Nicolas Klodt, Pascal Lenzner, Jonas Schmidt, George Skretas, and Armin Wells. How to reduce temporal cliques to find sparse spanners. In 32nd Annual European Symposium on Algorithms ( ESA ) , volume 308 of LIPIcs , pages 11:1--11:15. Schloss Dagstuhl - Leibniz-Zentrum f \" u ...

work page 2024

[4] [4]

On the size and the approximability of minimum temporally connected subgraphs

Kyriakos Axiotis and Dimitris Fotakis. On the size and the approximability of minimum temporally connected subgraphs. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP) , pages 149:1--149:14, 2016

work page 2016

[5] [5]

A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs

Surender Baswana and Sandeep Sen. A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. Random Structures & Algorithms , 30(4):532--563, 2007

work page 2007

[6] [6]

Sparse temporal spanners with low stretch

Davide Bil \`o , Gianlorenzo D'Angelo, Luciano Gual \`a , Stefano Leucci, and Mirko Rossi. Sparse temporal spanners with low stretch. In 30th Annual European Symposium on Algorithms (ESA) . Schloss Dagstuhl-Leibniz-Zentrum f \"u r Informatik, 2022

work page 2022

[7] [7]

Blackout-tolerant temporal spanners

Davide Bil \`o , Gianlorenzo D’Angelo, Luciano Gual \`a , Stefano Leucci, and Mirko Rossi. Blackout-tolerant temporal spanners. In International Symposium on Algorithms and Experiments for Wireless Sensor Networks , pages 31--44. Springer, 2022

work page 2022

[8] [8]

Temporal network creation games

Davide Bilò, Sarel Cohen, Tobias Friedrich, Hans Gawendowicz, Nicolas Klodt, Pascal Lenzner, and George Skretas. Temporal network creation games. In Edith Elkind, editor, Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence, IJCAI-23 , pages 2511--2519. International Joint Conferences on Artificial Intelligence Organi...

work page 2023