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On the Complexity of Secluded Path Problems

Authors Tesshu Hanaka , Daisuke Tsuru

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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Secluded path
  • Parameterized complexity
  • Polynomial-time algorithm

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Abstract

This paper investigates the complexity of finding secluded paths in graphs. We focus on the Short Secluded Path problem and a natural new variant we introduce, Shortest Secluded Path. Formally, given an undirected graph G = (V, E), two vertices s,t ∈ V, and two integers k,l, the 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 provide parameterized algorithms for the classic s-t k-Path problem. Furthermore, we introduce the 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.

Cite As Get BibTex

Tesshu Hanaka and Daisuke Tsuru. On the Complexity of Secluded Path Problems. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.IPEC.2025.4

Author Details

Tesshu Hanaka
  • Kyushu University, Fukuoka, Japan
Daisuke Tsuru
  • Kyushu University, Fukuoka, Japan

Funding

  • Hanaka, Tesshu: JSPS KAKENHI Grant Numbers JP21K17707, JP22H00513, JP25K03077, and JST, CRONOS, Japan Grant Number JPMJCS24K2.

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