Geodesic
Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1277-1291 Cet article a éte moissonné depuis la source Library of Science

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If G is a graph and X⊆ V(G), then X is a total mutual-visibility set if every pair of vertices x and y of G admits a shortest x,y-path P with V(P) ∩ X ⊆{x,y}. The cardinality of a largest total mutual-visibility set of G is the total mutual-visibility number μ_t(G) of G. Graphs with μ_t(G) = 0 are characterized as the graphs in which every vertex is the central vertex of a convex P_3. The total mutual-visibility number of Cartesian products is bounded and several exact results proved. For instance, μ_t(K_n □ K_m) = max{n,m} and μ_t(T □ H) = μ_t(T)μ_t(H), where T is a tree and H an arbitrary graph. It is also demonstrated that μ_t(G □ H) can be arbitrary larger than μ_t(G)μ_t(H).
Keywords: mutual-visibility set, total mutual-visibility set, bypass vertex, Cartesian product of graphs, tree 
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Tian, Jing; Klavžar, Sandi. Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1277-1291. http://geodesic.mathdoc.fr/item/DMGT_2024_44_4_a2/

[1] S. Cicerone, G. Di Stefano and S. Klavžar, On the mutual-visibility in Cartesian products and in triangle-free graphs, Appl. Math. Comput. 438 (2023) 127619. https://doi.org/10.1016/j.amc.2022.127619

[2] S. Cicerone, G. Di Stefano, S. Klavžar and I.G. Yero, Mutual-visibility in strong products of graphs via total mutual-visibility (2022). arXiv: 2210.07835

[3] G.A. Di Luna, P. Flocchini, S. Gan Chaudhuri, F. Poloni, N. Santoro and G. Viglietta, Mutual visibility by luminous robots without collisions, Inform. and Comput. 254 (2017) 392–418. https://doi.org/10.1016/j.ic.2016.09.005

[4] G. Di Stefano, Mutual visibility in graphs, Appl.\ Math. Comput. 419 (2022) 126850. https://doi.org/10.1016/j.amc.2021.126850

[5] P. Flocchini, G. Prencipe and N. Santoro, Distributed Computing by Oblivious Mobile Robots (Morgan & Claypool Publishers, 2012).

[6] M. Ghorbani, H.R. Maimani, M. Momeni, F. Rahimi-Mahid, S. Klavžar and G. Rus, The general position problem on Kneser graphs and on some graph operations, Discuss. Math. Graph Theory 41 (2021) 1199–1213. https://doi.org/10.7151/dmgt.2269

[7] W. Imrich, S. Klavžar and D.F. Rall, Topics in Graph Theory. Graphs and Their Cartesian Products (A K Peters/CRC Press, New York, 2008). https://doi.org/10.1201/b10613

[8] S. Klavžar, B. Patkós, G. Rus and I.G. Yero, On general position sets in Cartesian products, Results Math. 76(3) (2021) 123. https://doi.org/10.1007/s00025-021-01438-x

[9] S. Klavžar and G. Rus, The general position number of integer lattices, Appl. Math. Comput. 390 (2021) 125664. https://doi.org/10.1016/j.amc.2020.125664

[10] P. Manuel and S. Klavžar, A general position problem in graph theory, Bull. Aust. Math. Soc. 98 (2018) 177–187. https://doi.org/10.1017/S0004972718000473

[11] B. Patkós, On the general position problem on Kneser graphs, Ars Math. Contemp. 18 (2020) 273–280. https://doi.org/10.26493/1855-3974.1957.a0f

[12] P. Poudel, G. Sharma and A. Aljohani, Sublinear-time mutual visibility for fat oblivious robots, in: Proc. ICDCN'19, ACM (2019) 238–247. https://doi.org/10.1145/3288599.3288602

[13] J. Tian and K. Xu, The general position number of Cartesian products involving a factor with small diameter, Appl. Math. Comput. 403 (2021) 126206. https://doi.org/10.1016/j.amc.2021.126206

[14] J. Tian, K. Xu and S. Klavžar, The general position number of the Cartesian product of two trees, Bull. Aust. Math. Soc. 104 (2021) 1–10. https://doi.org/10.1017/S0004972720001276

[15] U. Chandran S.V. and G.J. Parthasarathy, The geodesic irredundant sets in graphs, Int. J. Math. Combin. 4 (2016) 135–143.

[16] D.B. West, Combinatorial Mathematics (Cambridge University Press, Cambridge, 2020).

[17] Y. Yao, M. He and S. Ji, On the general position number of two classes of graphs, Open Math. 20 (2022) 1021–1029. https://doi.org/10.1515/math-2022-0444

[18] K. Zarankiewicz, Problem P 101, Colloq.\ Math. 2 (1951) 301.

[19] M. Zhai, L. Fang and J. Shu, On the Turán number of theta graphs, Graphs Combin. 37 (2021) 2155–2165. https://doi.org/10.1007/s00373-021-02342-5