Abstract
We introduce a new model of indeterminacy in graphs: instead of specifying all the edges of the graph, the input contains all triples of vertices that form a connected subgraph. In general, different (labelled) graphs may have the same set of connected triples, making unique reconstruction of the original graph from the triples impossible. We identify some families of graphs (including triangle-free graphs) for which all graphs have a different set of connected triples. We also give algorithms that reconstruct a graph from a set of triples, and for testing if this reconstruction is unique. Finally, we study a possible extension of the model in which the subsets of size k that induce a connected graph are given for larger (fixed) values of k.
LC is supported by the Institute for Basic Science (IBS-R029-C1) and CG by Marie-Skłodowska Curie grant GRAPHCOSY (number 101063180). MvK and JV are supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 612.001.651.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Ahmed, M., Wenk, C.: Constructing street networks from GPS trajectories. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 60–71. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33090-2_7
Aspvall, B., Plass, M.F., Tarjan, R.E.: A linear-time algorithm for testing the truth of certain quantified Boolean formulas. Inf. Process. Lett. 8(3), 121–123 (1979)
Bastide, P., et al.: Reconstructing graphs from connected triples (2023)
Bollobás, B.: Random graphs. In: Modern Graph Theory. GTM, vol. 184, pp. 215–252. Springer, New York (1998). https://doi.org/10.1007/978-1-4612-0619-4_7
Bondy, J.A., Hemminger, R.L.: Graph reconstruction - a survey. J. Graph Theory 1(3), 227–268 (1977)
Bowler, A., Brown, P., Fenner, T.: Families of pairs of graphs with a large number of common cards. J. Graph Theory 63(2), 146–163 (2010)
Brandes, U., Cornelsen, S.: Phylogenetic graph models beyond trees. Discrete Appl. Math. 157(10), 2361–2369 (2009)
Cameron, P.J., Martins, C.: A theorem on reconstruction of random graphs. Comb. Probab. Comput. 2(1), 1–9 (1993)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, pp. 197–200. MIT press (2022)
Dey, T.K., Wang, J., Wang, Y.: Graph reconstruction by discrete Morse theory. In: Proceedings of the 34th International Symposium on Computational Geometry. Leibniz International Proceedings in Informatics (LIPIcs), vol. 99, pp. 31:1–31:15 (2018)
Even, S., Itai, A., Shamir, A.: On the complexity of timetable and multicommodity flow problems. SIAM J. Comput. 5(4), 691–703 (1976)
Feder, T.: Network flow and 2-satisfiability. Algorithmica 11(3), 291–319 (1994)
Fredman, M.L., Komlós, J., Szemerédi, E.: Storing a sparse table with \(0(1)\) worst case access time. J. ACM 31(3), 538–544 (1984)
Frieze, A., Karonski, M.: Introduction to Random Graphs. Cambridge University Press, New York (2015)
Giles, W.B.: The reconstruction of outerplanar graphs. J. Comb. Theory Ser. B 16(3), 215–226 (1974)
Groenland, C., Guggiari, H., Scott, A.: Size reconstructibility of graphs. J. Graph Theory 96(2), 326–337 (2021)
Harary, F.: A survey of the reconstruction conjecture. In: Bari, R.A., Harary, F. (eds.) Graphs and Combinatorics. LNCS, vol. 406, pp. 18–28. Springer, Berlin (1974). https://doi.org/10.1007/BFb0066431
Janson, S., Rucinski, A., Luczak, T.: Random Graphs. John Wiley & Sons, Hoboken (2011)
Kannan, S., Mathieu, C., Zhou, H.: Graph reconstruction and verification. ACM Trans. Algorithms 14(4), 1–30 (2018)
Kassiano, V., Gounaris, A., Papadopoulos, A.N., Tsichlas, K.: Mining uncertain graphs: an overview. In: Sellis, T., Oikonomou, K. (eds.) ALGOCLOUD 2016. LNCS, vol. 10230, pp. 87–116. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-57045-7_6
Kelly, P.J.: On Isometric Transformations. Ph.D. thesis, University of Wisconsin (1942)
Lauri, J.: The reconstruction of maximal planar graphs. J. Combi. Theory Ser. B 30(2), 196–214 (1981)
Lauri, J., Scapellato, R.: Topics in Graph Automorphisms and Reconstruction. Cambridge University Press, Cambridge (2016)
Mordeson, J.N., Peng, C.S.: Operations on fuzzy graphs. Inf. Sci. 79(3), 159–170 (1994)
Mossel, E., Ross, N.: Shotgun assembly of labeled graphs. IEEE Trans. Netw. Sci. Eng. 6(2), 145–157 (2017)
Myrvold, W.: The degree sequence is reconstructible from \(n- 1\) cards. Discrete Math. 102(2), 187–196 (1992)
Rosenfeld, A.: Fuzzy graphs. In: Zadeh, L.A., Fu, K.S., Tanaka, K., Shimura, M. (eds.) Fuzzy Sets and their Applications to Cognitive and Decision Processes, pp. 77–95. Elsevier (1975)
Tutte, W.: All the king’s horses. A guide to reconstruction. Graph Theory Relat. Top., 15–33 (1979)
Ulam, S.M.: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, vol. 8. Interscience Publishers (1960)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Bastide, P. et al. (2023). Reconstructing Graphs from Connected Triples. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_2
Download citation
DOI: https://doi.org/10.1007/978-3-031-43380-1_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-43379-5
Online ISBN: 978-3-031-43380-1
eBook Packages: Computer ScienceComputer Science (R0)