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Plane Strong Connectivity Augmentation

Authors Stéphane Bessy , Daniel Gonçalves , Amadeus Reinald , Dimitrios M. Thilikos

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Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Connectivity augmentation
  • Directed graphs
  • Parameterized complexity

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Abstract

We investigate the problem of strong connectivity augmentation within plane oriented graphs. We show that deciding whether a plane oriented graph D can be augmented with (any number of) arcs X such that D+X is strongly connected, but still plane and oriented, is NP-hard. The hardness also holds for the planar variant. This question becomes trivial within plane (or planar) digraphs, like most connectivity augmentation problems without a budget constraint. The budgeted variant, Plane Strong Connectivity Augmentation (PSCA) considers a plane oriented graph D along with some integer k, and asks for an X of size at most k ensuring that D+X is strongly connected, while remaining plane and oriented. Our main result is a fixed-parameter tractable algorithm for PSCA, running in time 2^O(k) n² log n. The cornerstone of our procedure is a structural result showing that, for any fixed k, each face admits a bounded number of partial solutions "dominating" all others. Then, our algorithm for PSCA combines face-wise branching with a randomized reduction to the polynomial Minimum Dijoin problem, yielding a Monte-Carlo FPT algorithm, which we derandomize. To the best of our knowledge, this is the first FPT algorithm for a (hard) connectivity augmentation problem constrained by planarity.

Cite As Get BibTex

Stéphane Bessy, Daniel Gonçalves, Amadeus Reinald, and Dimitrios M. Thilikos. Plane Strong Connectivity Augmentation. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 27:1-27:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.27

Author Details

Stéphane Bessy
  • LIRMM, Univ Montpellier, CNRS, Montpellier, France
Daniel Gonçalves
  • LIRMM, Univ Montpellier, CNRS, Montpellier, France
Amadeus Reinald
  • Institute of Informatics, University of Warsaw, Poland
  • LIRMM, Univ Montpellier, CNRS, Montpellier, France
Dimitrios M. Thilikos
  • LIRMM, Univ Montpellier, CNRS, Montpellier, France

Funding

SB and DT were supported by the ANR project GODASse ANR-24-CE48-4377.
  • Reinald, Amadeus: AR was supported by the Polish National Science Centre SONATA BIS-12 grant number 2022/46/E/ST6/00143.
  • Thilikos, Dimitrios M.: DT was also supported by the French-German Collaboration ANR/DFG Project UTMA (ANR-20-CE92-0027), the Franco-Norwegian project PHC AURORA 2024-2025 (Projet n° 51260WL), and the French National Research Agency (ANR) under the France 2030 grant reference number ANR-24-RRII-0002 operated by the Inria Quadrant Program.

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