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An Adjacency Labeling Scheme Based on a Decomposition of Trees into Caterpillars

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Combinatorial Algorithms (IWOCA 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13270))

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Abstract

In this paper we look at the problem of adjacency labeling of graphs. Given a family of undirected graphs the problem is to determine an encoding-decoding scheme for each member of the family such that we can decode the adjacency information of any pair of vertices only from their encoded labels. Further, we want the length of each label to be short (logarithmic in n, the number of vertices) and the encoding-decoding scheme to be computationally efficient. We propose a simple tree-decomposition based encoding scheme and use it give an adjacency labeling of size \(O(k \log k \log n)\)-bits. Here k is the clique-width of the graph family. We also extend the result to a certain family of k-probe graphs.

A. Banerjee–This research was supported in part by the DTIC contract FA8075-14-D-0002/0007.

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Notes

  1. 1.

    f(n) can be computed in time O(f(n)).

  2. 2.

    The vertex set of \(V(G_1\oplus G_2)\) of \(G_1\oplus G_2\) is \(V(G_1) \cup V(G_2)\) and the edge set \(E(G_1\oplus G_2) = E(G_1)\cup E(G_2)\).

  3. 3.

    It is a tree and not a DAG as the same graph does not take part in two separate union operations.

  4. 4.

    H is a spanning subgraph of G if \(V(H)=V(G)\) and \(E(H) \subseteq E(G)\).

  5. 5.

    Here \([n] = \{1,\ldots ,n\}\).

  6. 6.

    Some authors call them probe-k \(\mathcal F\)-graph [12].

  7. 7.

    We thank an anonymous reviewer for pointing this out.

  8. 8.

    Alternatively, we may assume that all relabeling operations in \(d_x\) proceed all join operations [13].

  9. 9.

    \([k] = \{1,\ldots ,k\}\).

References

  1. Abrahamsen, M., Alstrup, S., Holm, J., Knudsen, M.B.T., Stöckel, M.: Near-optimal induced universal graphs for bounded degree graphs. arXiv preprint arXiv:1607.04911 (2016)

  2. Alon, N.: Asymptotically optimal induced universal graphs. Geom. Funct. Anal. 27(1), 1–32 (2017). https://doi.org/10.1007/s00039-017-0396-9

    Article  MathSciNet  MATH  Google Scholar 

  3. Alstrup, S., Dahlgaard, S., Knudsen, M.B.T.: Optimal induced universal graphs and adjacency labeling for trees. J. ACM (JACM) 64(4), 1–22 (2017)

    Article  MathSciNet  Google Scholar 

  4. Alstrup, S., Dahlgaard, S., Knudsen, M.B.T., Porat, E.: Sublinear distance labeling. arXiv preprint arXiv:1507.02618 (2015)

  5. Alstrup, S., Kaplan, H., Thorup, M., Zwick, U.: Adjacency labeling schemes and induced-universal graphs. In: Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, pp. 625–634 (2015)

    Google Scholar 

  6. Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM (1999)

    Google Scholar 

  7. Breuer, M.A., Folkman, J.: An unexpected result in coding the vertices of a graph. J. Math. Anal. Appl. 20(3), 583–600 (1967)

    Article  MathSciNet  Google Scholar 

  8. Breuer, M.: Coding the vertexes of a graph. IEEE Trans. Inf. Theory 12(2), 148–153 (1966)

    Article  MathSciNet  Google Scholar 

  9. Chakraborty, S., Jo, S., Sadakane, K., Satti, S.R.: Succinct data structures for small clique-width graphs. In: 2021 Data Compression Conference (DCC), pp. 133–142. IEEE (2021)

    Google Scholar 

  10. Chandler, D.B., Chang, M.S., Kloks, T., Liu, J., Peng, S.L.: Partitioned probe comparability graphs. Theoret. Comput. Sci. 396(1–3), 212–222 (2008)

    Article  MathSciNet  Google Scholar 

  11. Chandler, D.B., Chang, M.S., Kloks, T., Liu, J., Peng, S.L.: On probe permutation graphs. Discret. Appl. Math. 157(12), 2611–2619 (2009)

    Article  MathSciNet  Google Scholar 

  12. Chang, M.S., Hung, L.J., Kloks, T., Peng, S.L.: Block-graph width. Theoret. Comput. Sci. 412(23), 2496–2502 (2011)

    Article  MathSciNet  Google Scholar 

  13. Courcelle, B., Engelfriet, J.: Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, vol. 138. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  14. Courcelle, B., Engelfriet, J., Rozenberg, G.: Handle-rewriting hypergraph grammars. J. Comput. Syst. Sci. 46(2), 218–270 (1993)

    Article  MathSciNet  Google Scholar 

  15. Courcelle, B., Vanicat, R.: Query efficient implementation of graphs of bounded clique-width. Discret. Appl. Math. 131(1), 129–150 (2003)

    Article  MathSciNet  Google Scholar 

  16. Dujmović, V., Esperet, L., Gavoille, C., Joret, G., Micek, P., Morin, P.: Adjacency labelling for planar graphs (and beyond). In: 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pp. 577–588. IEEE (2020)

    Google Scholar 

  17. Gavoille, C., Labourel, A.: Shorter implicit representation for planar graphs and bounded treewidth graphs. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 582–593. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-75520-3_52

    Chapter  Google Scholar 

  18. Hung, L.-J., Kloks, T.: On some simple widths. In: Rahman, M.S., Fujita, S. (eds.) WALCOM 2010. LNCS, vol. 5942, pp. 204–215. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11440-3_19

    Chapter  Google Scholar 

  19. Kamali, S.: Compact representation of graphs of small clique-width. Algorithmica 80(7), 2106–2131 (2018)

    Article  MathSciNet  Google Scholar 

  20. Kannan, S., Naor, M., Rudich, S.: Implicat representation of graphs. SIAM J. Discret. Math. 5(4), 596–603 (1992)

    Article  Google Scholar 

  21. Meer, K., Rautenbach, D.: On the OBDD size for graphs of bounded tree-and clique-width. Discret. Math. 309(4), 843–851 (2009)

    Article  MathSciNet  Google Scholar 

  22. Miller, G.L., Reif, J.H.: Parallel tree contraction and its application. Technical report, Harvard Univ. Cambridge, MA, Aiken Computation LAB (1985)

    Google Scholar 

  23. Muller, J.H.: Local structure in graph classes (1989)

    Google Scholar 

  24. Scheinerman, E.: Efficient local representations of graphs. In: Gera, R., Hedetniemi, S., Larson, C. (eds.) Graph Theory. PBM, pp. 83–94. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-31940-7_6

    Chapter  Google Scholar 

  25. Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26(3), 362–391 (1983)

    Article  MathSciNet  Google Scholar 

  26. Spinrad, J.: Efficient Graph Representations. Fields Institute Monographs, vol. 19. American Mathematical Society (2003)

    Google Scholar 

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Acknowledgement

The author would like to thank Guoli Ding for many discussions and considerable advice. In particular for the proof of caterpillar decomposition. We also thank anonymous reviewers for their helpful comments.

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  1. Missouri S&T, Rolla, MO, 65401, USA

    Avah Banerjee

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  1. Avah Banerjee
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Correspondence to Avah Banerjee .

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  1. Université Paris-Dauphine, Paris, France

    Cristina Bazgan

  2. Universität Trier, Trier, Germany

    Henning Fernau

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Banerjee, A. (2022). An Adjacency Labeling Scheme Based on a Decomposition of Trees into Caterpillars. In: Bazgan, C., Fernau, H. (eds) Combinatorial Algorithms. IWOCA 2022. Lecture Notes in Computer Science, vol 13270. Springer, Cham. https://doi.org/10.1007/978-3-031-06678-8_9

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  • DOI: https://doi.org/10.1007/978-3-031-06678-8_9

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