pith. sign in

arxiv: 2605.15833 · v1 · pith:SG6M2SEAnew · submitted 2026-05-15 · 💻 cs.DS · cs.DM· math.CO

Exploration of k-edge-deficient temporal graphs in linear time

Ivan Lahtin , Viktor Zamaraev This is my paper

Pith reviewed 2026-05-19 19:15 UTC · model grok-4.3

classification 💻 cs.DS cs.DMmath.CO
keywords temporal explorationtemporal graphsk-edge-deficient graphsalways-connected temporal graphsgraph traversallinear-time algorithmspolynomial-time algorithms
0
0 comments X

The pith

Any always-connected k-edge-deficient temporal graph admits an exploration schedule of length O(nk log k) computable in polynomial time.

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

The paper shows that temporal graphs in which every snapshot is connected and each differs from one fixed underlying n-vertex graph by at most k edges can be explored in O(nk log k) steps. This bound is linear in the number of vertices once k is fixed and improves prior O(kn log n) results by removing the dependence on log n. A sympathetic reader cares because the result indicates that when a temporal graph stays structurally close to a static one, its exploration time stays close to the linear time required for static graphs. The construction also yields a polynomial-time algorithm that produces the schedule.

Core claim

We resolve this question by showing that any always-connected k-edge-deficient temporal graph admits an exploration schedule of length O(nk log k). Moreover, given such a temporal graph, the corresponding exploration schedule can be computed in polynomial time. The obtained bound is linear in the number of vertices up to a factor depending only on k, removes the extraneous logarithmic dependence on n, and is nearly optimal. In particular, for constant k, our result yields an order-optimal Theta(n) exploration time, showing that temporal exploration in this near-static regime essentially retains the linear-time character of static graph traversal.

What carries the argument

A polynomial-time algorithm that builds an exploration schedule whose length is bounded by O(nk log k) for any always-connected k-edge-deficient temporal graph.

If this is right

  • For any fixed k the exploration length is Theta(n), matching the order of static-graph traversal.
  • The upper bound no longer carries an extra logarithmic factor in the number of vertices.
  • The schedule remains polynomial-time constructible, so the result is algorithmic as well as existential.
  • The bound is tight up to constants once k is fixed, given the known Omega(n log k) lower bound.

Where Pith is reading between the lines

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

  • Small bounded changes to an otherwise static graph do not force asymptotically slower exploration than the static case.
  • The same style of argument may extend to temporal graphs whose snapshots differ by a slowly growing function of n rather than a fixed k.
  • One could test whether the log k factor can be improved by constructing explicit worst-case instances for small k.

Load-bearing premise

Every snapshot must be connected and must differ from the same fixed underlying n-vertex graph by at most k edges.

What would settle it

A concrete family of always-connected k-edge-deficient temporal graphs whose shortest possible exploration schedule exceeds any constant multiple of nk log k would show the claimed bound does not hold.

read the original abstract

We study the Temporal Exploration problem, where an agent must visit all vertices of a temporal graph while traversing at most one available edge per time step. Unlike static graphs, which can be explored in linear time, temporal constraints can substantially increase exploration time even when every snapshot of the graph is connected. To better understand the source of this complexity, we focus on a near-static setting and consider always-connected $k$-edge-deficient temporal graphs, in which each snapshot is connected and differs from a fixed underlying $n$-vertex graph by at most $k$ edges. Although such graphs are structurally close to static graphs, they can still exhibit non-trivial temporal behaviour. Prior work showed that these graphs can be explored in $O(kn \log n)$ time steps and established a lower bound of $\Omega(n \log k)$, leaving open whether linear-time exploration in $n$ is possible. We resolve this question by showing that any always-connected $k$-edge-deficient temporal graph admits an exploration schedule of length $O(nk \log k)$. Moreover, given such a temporal graph, the corresponding exploration schedule can be computed in polynomial time. The obtained bound is linear in the number of vertices up to a factor depending only on $k$, removes the extraneous logarithmic dependence on $n$, and is nearly optimal. In particular, for constant $k$, our result yields an order-optimal $\Theta(n)$ exploration time, showing that temporal exploration in this near-static regime essentially retains the linear-time character of static graph traversal.

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

2 major / 3 minor

Summary. The paper studies the Temporal Exploration problem on always-connected k-edge-deficient temporal graphs, where each snapshot is connected and differs from a fixed underlying n-vertex graph by at most k edges. It claims that any such graph admits an exploration schedule of length O(nk log k) that can be computed in polynomial time. This improves the prior O(kn log n) upper bound, nearly matches the Omega(n log k) lower bound, and yields Theta(n) exploration time for constant k.

Significance. If the result holds, it resolves the open question of whether linear-in-n exploration is possible for fixed k in this near-static regime. The constructive polynomial-time algorithm is a strength, as is the removal of the extraneous log n factor while retaining only a k-dependent factor. This shows that temporal exploration in k-edge-deficient graphs essentially retains the linear-time character of static graph traversal.

major comments (2)
  1. [§3] §3, main theorem: the O(nk log k) bound is derived via a recursive decomposition that handles edge deficiencies by doubling or layering; the analysis must explicitly verify that the always-connected property is maintained across recursive calls on subgraphs, otherwise the induction may not go through.
  2. [Theorem 4.2] Theorem 4.2 (polynomial-time computation): the running time of the schedule construction algorithm is stated as polynomial but the degree is not specified; if it is O(n^3 k^2) or higher this could affect practicality, and a tighter bound or implementation detail would strengthen the claim.
minor comments (3)
  1. [Abstract] Abstract: the phrase 'linear time' is used but the bound is O(nk log k); for clarity, state explicitly that the result is linear in n for any fixed k.
  2. [Introduction] Introduction: the prior O(kn log n) result and Omega(n log k) lower bound should be cited with specific paper references rather than described generically.
  3. [§2] Notation: the definition of k-edge-deficient should be restated with a formal equation (e.g., |E_t Δ E_fixed| ≤ k for each time t) to avoid ambiguity in later sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment and constructive feedback on our manuscript. We address the major comments point by point below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

  1. Referee: [§3] §3, main theorem: the O(nk log k) bound is derived via a recursive decomposition that handles edge deficiencies by doubling or layering; the analysis must explicitly verify that the always-connected property is maintained across recursive calls on subgraphs, otherwise the induction may not go through.

    Authors: We agree with this observation. In the proof of the main theorem in Section 3, the recursive decomposition is applied to sub-instances that inherit the always-connected k-edge-deficient property from the original temporal graph. To make this explicit, we will add a short lemma in the revised manuscript establishing that the subgraphs considered in the recursion remain always-connected and k-edge-deficient. This ensures the induction goes through without issues. revision: yes

  2. Referee: [Theorem 4.2] Theorem 4.2 (polynomial-time computation): the running time of the schedule construction algorithm is stated as polynomial but the degree is not specified; if it is O(n^3 k^2) or higher this could affect practicality, and a tighter bound or implementation detail would strengthen the claim.

    Authors: We thank the referee for pointing this out. We will revise Theorem 4.2 to include an explicit bound on the running time of the algorithm. The schedule construction proceeds via a recursive decomposition with a logarithmic number of layers in k, and each step involves standard graph algorithms that can be implemented in polynomial time; we will provide the detailed complexity analysis in the revision to address practicality concerns. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper advances a constructive polynomial-time algorithm establishing an O(nk log k) exploration schedule for always-connected k-edge-deficient temporal graphs, directly resolving the open question left by the prior O(kn log n) upper bound and Omega(n log k) lower bound. No load-bearing step reduces by definition or construction to fitted parameters, self-referential inputs, or a self-citation chain; the result follows from the structural assumptions on the temporal graph without renaming known patterns or smuggling ansatzes. The derivation remains independent of the target bound itself and is externally falsifiable via the stated always-connected and k-edge-deficient conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper introduces no free parameters, new axioms, or invented entities; the result is a constructive algorithmic bound for the defined class of graphs using standard graph-theoretic assumptions.

pith-pipeline@v0.9.0 · 5808 in / 1194 out tokens · 69926 ms · 2026-05-19T19:15:53.094372+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
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.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    International Journal of Parallel, Emergent and Distributed Systems , volume =

    Casteigts, Arnaud and Flocchini, Paola and Quattrociocchi, Walter and Santoro, Nicola , title =. International Journal of Parallel, Emergent and Distributed Systems , volume =. 2012 , doi =

    work page 2012

  2. [2]

    Internet Mathematics , volume =

    Michail, Othon , title =. Internet Mathematics , volume =. 2016 , doi =

    work page 2016

  3. [3]

    , title =

    Michail, Othon and Spirakis, Paul G. , title =. Theoretical Computer Science , volume =. 2016 , doi =

    work page 2016

  4. [4]

    Journal of Computer and System Sciences , volume =

    Erlebach, Thomas and Hoffmann, Michael and Kammer, Frank , title =. Journal of Computer and System Sciences , volume =. 2021 , doi =

    work page 2021

  5. [5]

    and van der Zanden, Tom C

    Bodlaender, Hans L. and van der Zanden, Tom C. , title =. Information Processing Letters , volume =. 2019 , doi =

    work page 2019

  6. [6]

    43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018) , year=

    Faster Exploration of Degree-Bounded Temporal Graphs , author=. 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018) , year=

    work page 2018

  7. [7]

    46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) , pages=

    Two moves per time step make a difference , author=. 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) , pages=

    work page 2019

  8. [8]

    1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022) , pages=

    Faster exploration of some temporal graphs , author=. 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022) , pages=

    work page 2022

  9. [9]

    , title =

    Erlebach, Thomas and Spooner, Jakob T. , title =. Acta Informatica , volume =. 2022 , doi =

    work page 2022

  10. [10]

    Journal of Computer and System Sciences , volume=

    Parameterised temporal exploration problems , author=. Journal of Computer and System Sciences , volume=. 2023 , publisher=

    work page 2023

  11. [11]

    18th International Symposium on Parameterized and Exact Computation (IPEC 2023) , year=

    Kernelizing Temporal Exploration Problems , author=. 18th International Symposium on Parameterized and Exact Computation (IPEC 2023) , year=

    work page 2023

  12. [12]

    51st International Colloquium on Automata, Languages, and Programming (ICALP 2024) , pages=

    Exploiting Automorphisms of Temporal Graphs for Fast Exploration and Rendezvous , author=. 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024) , pages=

    work page 2024

  13. [13]

    arXiv preprint arXiv:2511.22604 , year=

    Improved exploration of temporal graphs , author=. arXiv preprint arXiv:2511.22604 , year=

    work page arXiv

  14. [14]

    Physics reports , volume=

    Temporal networks , author=. Physics reports , volume=. 2012 , publisher=

    work page 2012

  15. [15]

    2019 , publisher=

    Temporal network theory , author=. 2019 , publisher=

    work page 2019

  16. [16]

    Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2026) , pages=

    Temporal exploration of random spanning tree models , author=. Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2026) , pages=

    work page 2026

  17. [17]

    2020 , school =

    Exploring Temporal Cycles and Grids , author =. 2020 , school =

    work page 2020

  18. [18]

    21st International Colloquium on Structural Information and Communication Complexity (SIROCCO 2014) , pages=

    Exploration of constantly connected dynamic graphs based on cactuses , author=. 21st International Colloquium on Structural Information and Communication Complexity (SIROCCO 2014) , pages=. 2014 , organization=

    work page 2014

  19. [19]

    International Journal of Foundations of Computer Science , volume=

    Computing shortest, fastest, and foremost journeys in dynamic networks , author=. International Journal of Foundations of Computer Science , volume=. 2003 , publisher=

    work page 2003