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Testing Whether a Subgraph Is Convex or Isometric

Author Sergio Cabello

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • convex subgraph
  • isometric subgraph
  • plane graph

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Abstract

We consider the following two algorithmic problems: given a graph G and a subgraph H ⊆ G, decide whether H is an isometric or a geodesically convex subgraph of G. It is relatively easy to see that the problems can be solved by computing the distances between all pairs of vertices. We provide a conditional lower bound showing that, for sparse graphs with n vertices and Θ(n) edges, we cannot expect to solve the problem in O(n^{2-ε}) time for any constant ε > 0. We also show that the problem can be solved in subquadratic time for planar graphs and in near-linear time for graphs of bounded treewidth. Finally, we provide a near-linear time algorithm for the setting where G is a plane graph and H is defined by a few cycles in G.

Cite As Get BibTex

Sergio Cabello. Testing Whether a Subgraph Is Convex or Isometric. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.12

Author Details

Sergio Cabello
  • Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
  • Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

Funding

Funded in part by the Slovenian Research and Innovation Agency (P1-0297, N1-0218, N1-0285). Funded in part by the European Union (ERC, KARST, project number 101071836). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

Acknowledgements

I am very grateful to Vladimir Batagelj and Sandi Klavžar for discussing the closest related work. I am also very grateful to a reviewer that provided several relevant references.

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