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Isometric-Universal Graphs for Trees

Authors Edgar Baucher , François Dross , Cyril Gavoille

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  • Theory of computation → Design and analysis of algorithms
Keywords
  • tree
  • forest
  • isometric subgraph
  • universal graph
  • distance-preserving

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Abstract

We consider the problem of finding the smallest graph that contains two input trees each with at most n vertices preserving their distances. In other words, we look for an isometric-universal graph with the minimum number of vertices for two given trees. We prove that this problem can be solved in time O(n^{5/2}log{n}). We extend this result to forests instead of trees, and propose an algorithm with running time O(n^{7/2}log{n}). As a key ingredient, we show that a smallest isometric-universal graph of two trees essentially is a tree. Furthermore, we prove that these results cannot be extended. Firstly, we show that deciding whether there exists an isometric-universal graph with t vertices for three forests is NP-complete. Secondly, we show that any smallest isometric-universal graph cannot be a tree for some families of three trees. This latter result has implications for greedy strategies solving the smallest isometric-universal graph problem.

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Edgar Baucher, François Dross, and Cyril Gavoille. Isometric-Universal Graphs for Trees. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.16

Author Details

Edgar Baucher
  • LaBRI, University of Bordeaux, CNRS, Talence, France
François Dross
  • LaBRI, University of Bordeaux, CNRS, Talence, France
Cyril Gavoille
  • LaBRI, University of Bordeaux, CNRS, Talence, France

References

  1. Faisal N. Abu-Khzam. Maximum common induced subgraph parameterized by vertex cover. Information Processing Letters, 114(3):99-103, March 2014. URL: https://doi.org/10.1016/j.ipl.2013.11.007.
  2. Faisal N. Abu-Khzam, Édouard Bonnet, and Florian Sikora. On the complexity of various parameterizations of common induced subgraph isomorphism. Theoretical Computer Science, 697:69-78, October 2017. URL: https://doi.org/10.1016/j.tcs.2017.07.010.
  3. Edgar Baucher, François Dross, and Cyril Gavoille. Isometric-universal graphs for trees. Technical Report 2506.11704v1 [cs.DS], arXiv, June 2025. URL: https://doi.org/10.48550/arXiv.2506.11704.
  4. Helena Bergold, Vesna Iršič, Robert Lauff, Joachim Orthaber, Manfred Scheucher, and Alexandra Wesolek. Subgraph-universal planar graphs for trees. Technical Report 2409.01678 [quant-ph], arXiv, September 2024. URL: https://doi.org/10.48550/arXiv.2409.01678.
  5. Olivier Bodini. On the minimum size of a contraction-universal tree. In 28th International Workshop on Graph-Theoretic Concepts in Computer Science (WG), volume 2573 of Lecture Notes in Computer Science, pages 25-34. Springer, June 2002. URL: https://doi.org/10.1007/3-540-36379-3_3.
  6. Gregory Bodwin. New results on linear size distance preservers. SIAM Journal on Computing, 50(2):662-673, January 2021. URL: https://doi.org/10.1137/19M123662X.
  7. Béla Bollobás, Don Coppersmith, and Michael Elkin. Sparse distance preservers and additive spanners. SIAM Journal on Discrete Mathematics, 19(4):1029-1055, 2006. URL: https://doi.org/10.1137/S0895480103431046.
  8. Horst Bunke, Xiaoyi Jiang, and Abraham Kandel. On the minimum common supergraph of two graphs. Computing, 65:13-25, July 2000. URL: https://doi.org/10.1007/PL00021410.
  9. Fan R.K. Chung, Ronald Lewis Graham, and J. Shearer. Note: Universal caterpillars. Journal of Combinatorial Theory, Series B, 31(3):348-355, December 1981. URL: https://doi.org/10.1016/0095-8956(81)90037-X.
  10. Don Coppersmith and Michael Elkin. Sparse source-wise and pair-wise distance preservers. SIAM Journal on Discrete Mathematics, 20(2):463-501, 2006. URL: https://doi.org/10.1137/S0895480104445319.
  11. Vida Dujmović, Louis Esperet, Cyril Gavoille, Gwenaël Joret, Piotr Micek, and Pat Morin. Adjacency labelling for planar graphs (and beyond). Journal of the ACM, 68(6):Article No. 42, pp. 1-33, December 2021. URL: https://doi.org/10.1145/3477542.
  12. Louis Esperet, Cyril Gavoille, and Carla Groenland. Isometric universal graphs. SIAM Journal on Discrete Mathematics, 5(2):1224-1237, 2021. URL: https://doi.org/10.1137/21M1406155.
  13. Louis Esperet, Gwenaël Joret, and Pat Morin. Sparse universal graphs for planarity. Journal of the London Mathematical Society, 108(4):1333-1357, October 2023. URL: https://doi.org/10.1112/jlms.12781.
  14. Michael R. Garey and David S. Johnson. Computers and Intractability - A Guide to the Theory of NP-Completeness. W.H. Freeman, 1979. Google Scholar
  15. Cyril Gavoille and Amaury Jacques. Compact universal graphs for small families of graphs, 2024. In preparation. Google Scholar
  16. Paweł Gawrychowski, Fabian Kuhn, Jakub 0.85opuszański, Konstantinos Panagiotou, and Pascal Su. Labeling schemes for nearest common ancestors through minor-universal trees. In 29th Symposium on Discrete Algorithms (SODA), pages 2604-2619. ACM-SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.166.
  17. Arvind Gupta and Naomi Nishimura. Finding largest subtrees and smallest supertrees. Algorithmica, 21(2):183-210, June 1998. URL: https://doi.org/10.1007/PL00009212.
  18. Dieter Rautenbach and Florian Werner. Induced subforests and superforests. Technical Report 2403.14492v1 [cs.DS], arXiv, March 2024. URL: https://doi.org/10.48550/arXiv.2403.14492.
  19. Peter Winkler. Proof of the squashed cube conjecture. Combinatorica, 3(1):135-139, March 1983. URL: https://doi.org/10.1007/BF02579350.
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