Abstract
A representation of an n-vertex graph G is implicit if it assigns to each vertex of G a binary code of length \(O(\log n)\) so that the adjacency of two vertices is a function of their codes. A necessary condition for a hereditary class \(\mathcal X\) of graphs to admit an implicit representation is that \(\mathcal X\) has at most factorial speed of growth. This condition, however, is not sufficient, as was recently shown in [10]. Several sufficient conditions for the existence of implicit representations deal with boundedness of some parameters, such as degeneracy or clique-width. In the present paper, we analyse more graph parameters and prove a number of new results related to implicit representation and factorial properties.
This work was supported by the Russian Science Foundation Grant No. 21-11-00194.
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References
Alecu, B., Atminas, A., Lozin, V.: Graph functionality. J. Combin. Theory Ser. B 147, 139–158 (2021). https://doi.org/10.1007/978-3-030-30786-8_11
Allen, P.: Forbidden induced bipartite graphs. J. Graph Theory 60, 219–241 (2009)
Atminas, A., Collins, A., Lozin, V., Zamaraev, V.: Implicit representations and factorial properties of graphs. Discrete Math. 338, 164–179 (2015)
Atminas, A.: Classes of graphs without star forests and related graphs. arXiv preprint arXiv:1711.01483
Balogh, J., Bollobás, B., Weinreich, D.: The speed of hereditary properties of graphs. J. Combin. Theory Ser. B 79, 131–156 (2000)
Bonnet, É., Geniet, C., Kim, E.J., Thomassé, S., Watrigant, R.: Twin-width II: small classes. SODA 2021: 1977–1996
Boros, E., Gurvich, V., Milanic, M.: Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1-Sperner hypergraphs. J. Graph Theory 94, 364–397 (2020)
Harms, N.: Universal communication, universal graphs, and graph labeling, ITCS (2020)
Harms, N., Wild, S., Zamaraev, V.: Randomized communication and implicit graph representations. STOC (2022)
Hatami, H., Hatami, P.: The implicit graph conjecture is false. arXiv preprint arXiv:2111.13198
Kannan, S., Naor, M., Rudich, S.: Implicit representation of graphs. SIAM J. Discrete Math. 5, 596–603 (1992)
Lozin, V.: Bipartite graphs without a skew star. Discrete Math. 257, 83–100 (2002)
Lozin, V., Zamaraev, V.: The structure and the number of \(P_7\)-free bipartite graphs. Eur. J. Comb. 65, 143–153 (2017)
Spinrad, J.P.: Efficient graph representations. Fields Institute Monographs, 19. American Mathematical Society, Providence, RI, pp. xiii+342 (2003)
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Alecu, B., Alekseev, V.E., Atminas, A., Lozin, V., Zamaraev, V. (2022). Graph Parameters, Implicit Representations and Factorial Properties. 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_5
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DOI: https://doi.org/10.1007/978-3-031-06678-8_5
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