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arxiv: 2603.22818 · v2 · submitted 2026-03-24 · 💻 cs.DS

On the Complexity of Secluded Path Problems

Tesshu Hanaka , Daisuke Tsuru This is my paper

Pith reviewed 2026-05-15 01:18 UTC · model grok-4.3

classification 💻 cs.DS
keywords secluded pathshortest pathparameterized complexitycliquewidthneighborhood diversitytwin coverdynamic programminggraph algorithms
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The pith

Shortest Secluded Path is solvable in polynomial time on unweighted graphs, and Short Secluded Path admits XP and FPT algorithms under specific width parameters.

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

The paper maps the complexity of locating s-t paths whose external neighbors are few. It supplies an XP algorithm for the Short Secluded Path decision problem when parameterized by cliquewidth and fixed-parameter algorithms when parameterized by neighborhood diversity or twin cover. These techniques also solve the classic s-t k-Path problem under the same parameters. The authors introduce the optimization version Shortest Secluded Path, which selects a shortest s-t path that minimizes the neighbor count, and prove this version lies in polynomial time on unweighted graphs while becoming W[1]-hard on weighted graphs yet remaining in XP when parameterized by distance.

Core claim

The authors establish that Short Secluded Path, which asks whether an s-t path of length at most k exists with at most l neighbors, is solvable in XP time parameterized by cliquewidth and in FPT time parameterized by neighborhood diversity or twin cover via dynamic programming. They further introduce Shortest Secluded Path, which seeks a shortest s-t path minimizing the number of external neighbors, and show that this problem admits a polynomial-time algorithm on unweighted graphs while it is W[1]-hard on edge-weighted graphs and lies in XP when parameterized by the shortest-path distance.

What carries the argument

Dynamic programming over decompositions according to cliquewidth, neighborhood diversity, and twin cover; for the unweighted optimization problem, a transformation that reduces the task to ordinary shortest-path computation while tracking external neighbors.

If this is right

  • The classic s-t k-Path problem becomes fixed-parameter tractable when parameterized by neighborhood diversity or twin cover.
  • Secluded-path queries become tractable on graphs whose vertices can be partitioned into few groups with identical neighborhoods.
  • On unweighted graphs the most secluded shortest route between any pair can be recovered by a standard polynomial-time shortest-path routine after a local modification.
  • When edge weights are present the problem remains solvable in XP time once the target distance is fixed as a parameter.

Where Pith is reading between the lines

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

  • The same width parameters may yield algorithms for seclusion-constrained versions of Steiner trees or disjoint paths.
  • Graphs with small neighborhood diversity, common in modular or cluster-structured networks, now admit efficient secluded-path search.
  • The polynomial-time result on unweighted graphs suggests that length and seclusion constraints can be optimized jointly without an inherent tradeoff when all edges have equal cost.
  • Practical implementations could test the new algorithms on social or infrastructure networks whose twin-cover or neighborhood-diversity values are modest.

Load-bearing premise

The algorithms assume that the input graphs admit standard decompositions with respect to the chosen parameters and that no extra structural restrictions beyond the parameter bounds are required for correctness.

What would settle it

An explicit unweighted graph with designated s and t on which any claimed polynomial-time procedure for Shortest Secluded Path returns a path whose external-neighbor count exceeds the true minimum among all shortest s-t paths.

Figures

Figures reproduced from arXiv: 2603.22818 by Daisuke Tsuru, Tesshu Hanaka.

Figure 1
Figure 1. Figure 1: An illustration of a 5-secluded s-t path P of length 5. The s-t path P consists of the red vertices. The blue vertices are neighbors of P [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Parameterized complexity of Short Secluded Path with respect to structural graph parameters. The connection between two parameters indicates that the upper parameter p is bounded by some computable function f(·) of the lower parameter q; that is, p ≤ f(q). Parameters marked with an asterisk (∗) represent our contributions in this paper. Double￾bordered rectangles indicate that the parameterized problem bel… view at source ↗
Figure 3
Figure 3. Figure 3: An illustration of a partition of V with respect to a path P in Claim 15. simply replace the clique C ′ used by P with C, that is, if P passes through c ′ vertices in C ′ between Pj and Pj+1, P ′ passes through arbitrary c ′ vertices in C between Pj and Pj+1. Since |C| ≥ |C ′ | ≥ c ′ , this replacement is possible. It is clear that |V (P ′ )| = |V (P)| holds. We show that P ′ satisfies |N(V (P))| = |N(V (P… view at source ↗
Figure 4
Figure 4. Figure 4: The reduction from Multicolored Clique to Shortest Secluded Path in The￾orem 22. The weight of each thin edge is 1 and the weight of each bold edge is k + 1. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
read the original abstract

This paper investigates the complexity of finding secluded paths in graphs. We focus on the \textsc{Short Secluded Path} problem and a natural new variant we introduce, \textsc{Shortest Secluded Path}. Formally, given an undirected graph $G=(V, E)$, two vertices $s,t\in V$, and two integers $k,l$, the \textsc{Short Secluded Path} problem asks whether there exists an $s$-$t$ path of length at most $k$ with at most $l$ neighbors. This problem is known to be computationally hard: it is W[1]-hard when parameterized by the path length $k$ or by cliquewidth, and para-NP-complete when parameterized by the number $l$ of neighbors. The fixed-parameter tractability is known for $k+l$ or treewidth. In this paper, we expand the parameterized complexity landscape by designing (1) an XP algorithm parameterized by cliquewidth and (2) fixed-parameter algorithms parameterized by neighborhood diversity and twin cover number, respectively. As a byproduct, our results also yield parameterized algorithms for the classic \textsc{$s$-$t$ $k$-Path} problem under the considered parameters. Furthermore, we introduce the \textsc{Shortest Secluded Path} problem, which seeks a shortest $s$-$t$ path with the minimum number of neighbors. In contrast to the hardness of the original problem, we reveal that this variant is solvable in polynomial time on unweighted graphs. We complete this by showing that for edge-weighted graphs, the problem becomes W[1]-hard yet remains in XP when parameterized by the shortest path distance between $s$ and $t$.

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 investigates the parameterized complexity of the Short Secluded Path problem (existence of an s-t path of length at most k with at most l external neighbors) and introduces the Shortest Secluded Path variant (a shortest s-t path minimizing the number of external neighbors). It gives an XP algorithm parameterized by cliquewidth, FPT algorithms parameterized by neighborhood diversity and by twin cover number, polynomial-time solvability for Shortest Secluded Path on unweighted graphs, and W[1]-hardness plus XP membership (by distance) for the weighted case; the results also apply to the classic s-t k-Path problem.

Significance. If the claims hold, the work meaningfully expands the parameterized complexity map for secluded-path problems by identifying new tractable parameters (neighborhood diversity and twin cover) beyond the known treewidth and k+l cases, while separating the decision and optimization variants via a clean polynomial-time result on unweighted graphs. The explicit dynamic-programming constructions over the respective decompositions and covers constitute a concrete algorithmic contribution.

minor comments (2)
  1. The abstract states that the Short Secluded Path problem is W[1]-hard parameterized by cliquewidth, but does not clarify whether this is for the single parameter or combined with k; a parenthetical reference to the relevant theorem would remove ambiguity.
  2. In the description of the DP for neighborhood diversity, the state component that tracks external neighbors is introduced without an explicit recurrence equation; adding a displayed recurrence (even if standard) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript. We appreciate the recognition that our XP algorithm for cliquewidth, the FPT results for neighborhood diversity and twin cover, the polynomial-time algorithm for Shortest Secluded Path on unweighted graphs, and the hardness/XP results for the weighted case meaningfully expand the parameterized complexity landscape for secluded-path problems and also apply to s-t k-Path.

Circularity Check

0 steps flagged

No significant circularity in algorithmic constructions

full rationale

The paper's central results consist of explicit XP and FPT algorithms constructed via standard dynamic programming over cliquewidth, neighborhood diversity, and twin cover decompositions, plus a direct polynomial-time procedure for the shortest variant on unweighted graphs. These derivations rely on expressing path existence and external-neighbor counting inside the respective structural decompositions; no step reduces by definition to its own inputs, no parameter is fitted and then renamed as a prediction, and no load-bearing claim depends on a self-citation chain or imported uniqueness theorem. Hardness statements reference external prior work, while the new positive results are self-contained constructive proofs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard graph-theoretic definitions and known hardness results for secluded paths. No free parameters are fitted to data; the algorithms are exact and combinatorial.

axioms (2)
  • standard math Standard definitions of cliquewidth, neighborhood diversity, and twin cover as graph parameters.
    Invoked throughout the parameterized algorithms section.
  • domain assumption Known W[1]-hardness and para-NP-completeness results for Short Secluded Path under the listed parameters.
    Used as baseline to position the new algorithms.

pith-pipeline@v0.9.0 · 5608 in / 1386 out tokens · 29527 ms · 2026-05-15T01:18:22.513256+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    R. Bellman. On a routing problem.Quarterly of Applied Mathematics, 16:87–90, 1958

    work page 1958

  2. [2]

    An optimal XP algorithm for hamiltonian cycle on graphs of bounded clique-width.Algorithmica, 82(6):1654–1674, 2020

    Benjamin Bergougnoux, Mamadou Moustapha Kanté, and O-joung Kwon. An optimal XP algorithm for hamiltonian cycle on graphs of bounded clique-width.Algorithmica, 82(6):1654–1674, 2020

    work page 2020

  3. [3]

    Johnson, Merav Parter, and David Peleg

    Shiri Chechik, Matthew P. Johnson, Merav Parter, and David Peleg. Secluded connectivity problems.Algorithmica, 79(3):708–741, 2017

    work page 2017

  4. [4]

    Cherkassky, Andrew V

    Boris V. Cherkassky, Andrew V. Goldberg, and Tomasz Radzik. Shortest paths algorithms: Theory and experimental evaluation.Math. Program., 73(2):129–174, May 1996

    work page 1996

  5. [5]

    Upper bounds to the clique width of graphs.Discret

    Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs.Discret. Appl. Math., 101(1-3):77–114, 2000

    work page 2000

  6. [6]

    Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh.Parameterized Algorithms

    Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh.Parameterized Algorithms. Springer, 2015

    work page 2015

  7. [7]

    Alexsander Andrade de Melo, Celina M. H. de Figueiredo, and Uéverton S. Souza. On the computational difficulty of the terminal connection problem.RAIRO Theor. Informatics Appl., 57:3, 2023

    work page 2023

  8. [8]

    E. W. Dijkstra. A note on two problems in connexion with graphs.Numer. Math., 1(1):269–271, December 1959

    work page 1959

  9. [9]

    Dijkstra

    Edsger W. Dijkstra. A note on two problems in connexion with graphs.Numerische Mathematik, 1:269–271, 1959

    work page 1959

  10. [10]

    Huib Donkers, Bart M. P. Jansen, and Jari J. H. de Kroon. Findingk-secluded trees faster. J. Comput. Syst. Sci., 138:103461, 2023

    work page 2023

  11. [11]

    Finding thekshortest paths.SIAM J

    David Eppstein. Finding thekshortest paths.SIAM J. Comput., 28(2):652–673, 1998. 21

    work page 1998

  12. [12]

    How to solve np-hard graph problems on clique-width bounded graphs in polynomial time

    Wolfgang Espelage, Frank Gurski, and Egon Wanke. How to solve np-hard graph problems on clique-width bounded graphs in polynomial time. In Andreas Brandstädt and Van Bang Le, editors,Graph-Theoretic Concepts in Computer Science, 27th International Workshop, WG 2001, Boltenhagen, Germany, June 14-16, 2001, Proceedings, volume 2204 ofLecture Notes in Comput...

    work page 2001

  13. [13]

    Fellows, Danny Hermelin, Frances A

    Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems.Theor. Comput. Sci., 410(1):53–61, 2009

    work page 2009

  14. [14]

    Fomin, Petr A

    Fedor V. Fomin, Petr A. Golovach, Nikolay Karpov, and Alexander S. Kulikov. Parameter- ized complexity of secluded connectivity problems.Theory Comput. Syst., 61(3):795–819, 2017

    work page 2017

  15. [15]

    Fomin, Petr A

    Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Intractability of clique-width parameterizations.SIAM J. Comput., 39(5):1941–1956, 2010

    work page 1941

  16. [16]

    An application of simultaneous diophantine approximation in combinatorial optimization.Comb., 7(1):49–65, 1987

    András Frank and Éva Tardos. An application of simultaneous diophantine approximation in combinatorial optimization.Comb., 7(1):49–65, 1987

    work page 1987

  17. [17]

    Parameterized algorithms for modular-width

    Jakub Gajarský, Michael Lampis, and Sebastian Ordyniak. Parameterized algorithms for modular-width. In Gregory Z. Gutin and Stefan Szeider, editors,Parameterized and Ex- act Computation - 8th International Symposium, IPEC 2013, Sophia Antipolis, France, September 4-6, 2013, Revised Selected Papers, volume 8246 ofLecture Notes in Computer Science, pages 16...

    work page 2013

  18. [18]

    Improving vertex cover as a graph parameter.Discret

    Robert Ganian. Improving vertex cover as a graph parameter.Discret. Math. Theor. Comput. Sci., 17(2):77–100, 2015

    work page 2015

  19. [19]

    Golovach, Pinar Heggernes, Paloma T

    Petr A. Golovach, Pinar Heggernes, Paloma T. Lima, and Pedro Montealegre. Finding connected secluded subgraphs.J. Comput. Syst. Sci., 113:101–124, 2020

    work page 2020

  20. [20]

    Finding a minimum spanning tree with a small non-terminal set.Theor

    Tesshu Hanaka and Yasuaki Kobayashi. Finding a minimum spanning tree with a small non-terminal set.Theor. Comput. Sci., 1033:115092, 2025

    work page 2025

  21. [21]

    On the complexity of secluded path problems

    Tesshu Hanaka and Daisuke Tsuru. On the complexity of secluded path problems. In Akanksha Agrawal and Erik Jan van Leeuwen, editors,20th International Symposium on Parameterized and Exact Computation, IPEC 2025, Warsaw, Poland, September 17-19, 2025, volume 358 ofLIPIcs, pages 4:1–4:16. Schloss Dagstuhl - Leibniz-Zentrum für Infor- matik, 2025

    work page 2025

  22. [22]

    Finding branch-decompositions and rank-decompositions

    Petr Hlinený and Sang-il Oum. Finding branch-decompositions and rank-decompositions. SIAM J. Comput., 38(3):1012–1032, 2008

    work page 2008

  23. [23]

    Lenstra Jr

    Hendrik W. Lenstra Jr. Integer programming with a fixed number of variables.Math. Oper. Res., 8(4):538–548, 1983

    work page 1983

  24. [24]

    Minkowski’s convex body theorem and integer programming.Math

    Ravi Kannan. Minkowski’s convex body theorem and integer programming.Math. Oper. Res., 12(3):415–440, 1987

    work page 1987

  25. [25]

    Algorithmic meta-theorems for restrictions of treewidth.Algorithmica, 64(1):19–37, 2012

    Michael Lampis. Algorithmic meta-theorems for restrictions of treewidth.Algorithmica, 64(1):19–37, 2012

    work page 2012

  26. [26]

    On the computational complexity of length- and neighborhood-constrained path problems.Inf

    Max-Jonathan Luckow and Till Fluschnik. On the computational complexity of length- and neighborhood-constrained path problems.Inf. Process. Lett., 156:105913, 2020. 22

    work page 2020

  27. [27]

    A parameterized study of secluded structures in directed graphs.CoRR, abs/2502.06048, 2025

    Nadym Mallek, Jonas Schmidt, and Shaily Verma. A parameterized study of secluded structures in directed graphs.CoRR, abs/2502.06048, 2025

    work page arXiv 2025

  28. [28]

    McConnell and Jeremy P

    Ross M. McConnell and Jeremy P. Spinrad. Modular decomposition and transitive orien- tation.Discret. Math., 201(1-3):189–241, 1999

    work page 1999

  29. [29]

    Approximatingrank-widthandclique-widthquickly.ACM Trans

    Sang-ilOum. Approximatingrank-widthandclique-widthquickly.ACM Trans. Algorithms, 5(1):10:1–10:20, 2008

    work page 2008

  30. [30]

    Sang-il Oum and Paul D. Seymour. Approximating clique-width and branch-width.J. Comb. Theory B, 96(4):514–528, 2006

    work page 2006

  31. [31]

    Corneil, Michel Habib, and Christophe Paul

    Marc Tedder, Derek G. Corneil, Michel Habib, and Christophe Paul. Simpler linear- time modular decomposition via recursive factorizing permutations. In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors,Automata, Languages and Programming, 35th International Collo- quium, ICALP 2008, Rey...

    work page 2008

  32. [32]

    Mertzios, Hendrik Molter, Manuel Sorge, and Ondrej Suchý

    René van Bevern, Till Fluschnik, George B. Mertzios, Hendrik Molter, Manuel Sorge, and Ondrej Suchý. The parameterized complexity of finding secluded solutions to some classical optimization problems on graphs.Discret. Optim., 30:20–50, 2018

    work page 2018

  33. [33]

    Tsidulko

    René van Bevern, Till Fluschnik, and Oxana Yu. Tsidulko. Parameterized algorithms and data reduction for the short secludeds-t-path problem.Networks, 75(1):34–63, 2020. 23

    work page 2020