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The geodesic-transversal problem on graphs of diameter at most three | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 23 خرداد 1405 اصل مقاله (369.82 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.30949.2677 | ||
| نویسندگان | ||
| Anand M G* 1؛ T Jeba Raj2 | ||
| 1Department of Mathematics, Malankara Catholic College, Manonmaniam Sundaranar University, Tirunelveli 627012, Tamil Nadu, India | ||
| 2Department of Mathematics, Malankara Catholic College, Mariagiri, Kaliyakkavilai, Tamil Nadu-629153, India | ||
| چکیده | ||
| In graph theory, finding the minimal set of vertices with specific covering properties is important. A geodesic transversal of a graph $G$ is a set $S$ of vertices such that every maximal geodesic of $G$ includes at least one vertex from S. The minimum size of a geodesic transversal of $G$ is called geodesic transversal number, denoted as $gt(G)$. It helps in understanding how graphs navigate, how efficiently they communicate, and how to monitor networks. This work finds $gt(G)$ for all graphs with a diameter of 2 and expands the analysis to various classes of graphs with a diameter of 3. | ||
| کلیدواژهها | ||
| geodesic transversal number؛ complementary prism؛ bipartite graph؛ split graphs | ||
| مراجع | ||
|
1] A.M. Anto, R. Rajeshkumar, L.E. Preshiba, and V.M.M. Rose, Insights into IF-geodetic convexity in intuitionistic fuzzy graphs: Harnessing the IF-geodetic wiener index for global human trading analysis and IF-geodetic cover for gateway node identification, Symmetry 17 (2025), no. 8, 1277. https://doi.org/10.3390/sym17081277 [2] A. Bendali-Braham, N. Ikhlef-Eschouf, and M. Blidia, Some results on the bchromatic number in complementary prism graphs, RAIRO Oper. Res. 53 (2019), no. 4, 1187–1195. https://doi.org/10.1051/ro/2018054
[3] A.A. Bertossi, Dominating sets for split and bipartite graphs, Inform. Process. Lett. 19 (1984), no. 1, 37–40. https://doi.org/10.1016/0020-0190(84)90126-1
[4] B. Brešar, T. Kos, R. Krivoš-Belluš, and G. Semanišin, Hitting subgraphs in $P_4$-tidy graphs, Appl. Math. Comput. 352 (2019), 211–219. https://doi.org/10.1016/j.amc.2019.01.074
[5] D.M. Cardoso, P. Carvalho, M.A.A. de Freitas, and C.T. Vinagre, Spectra, signless Laplacian and Laplacian spectra of complementary prisms of graphs, Linear Algebra Appl. 544 (2018), 325–338. https://doi.org/10.1016/j.laa.2018.01.020
[6] U. Chandran S V, G. Di Stefano, E.J. Thomas, and J. Tuite, Colouring a graph with position sets, Ars Math. Contemp. (2025), In–press.
[7] M.A. Duarte, L. Penso, D. Rautenbach, and U. dos Santos Souza, Complexity properties of complementary prisms, J. Comb. Optim. 33 (2017), no. 2, 365–372. https://doi.org/10.1007/s10878-015-9968-5
8] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Company, San Francisco, 1979.
[9] M.R. Garey and R.E. Tarjan, A linear-time algorithm for finding all feedback vertices, Inform. Process. Lett. 7 (1978), no. 6, 274–276. https://doi.org/10.1016/0020-0190(78)90015-7
[10] F. Harary and R.A. Melter, On the metric dimension of a graph, Ars Combin. 2 (1976), 191–195.
[11] T. Haynes, M. Henning, P. Slater, and L. Merwe, The complementary product of two graphs, Bull. Inst. Combin. Appl. 51 (2007), 21–30.
[12] T.W. Haynes, M.A. Henning, and L.C. Van Der Merwe, Domination and total domination in complementary prisms, J. Comb. Optim. 18 (2009), no. 1, 23–37. https://doi.org/10.1007/s10878-007-9135-8
[13] T. James, S. Klavžar, and A. Vijayakumar, The domination game on split graphs, Bull. Aust. Math. Soc. 99 (2019), no. 2, 327–337. https://doi.org/10.1017/S0004972718001053
[14] D. Korže and A. Vesel, General position sets in two families of cartesian product graphs, Mediterr. J. Math. 20 (2023), no. 4, 203. https://doi.org/10.1007/s00009-023-02416-z
[15] P. Manuel, B. Brešar, and S. Klavžar, The geodesic-transversal problem, Appl. Math. Comput. 413 (2022), 126621. https://doi.org/10.1016/j.amc.2021.126621 [16] P. Manuel, B. Brešar, and S. Klavžar, Geodesic packing in graphs, Appl. Math. Comput. 459 (2023), 128277. https://doi.org/10.1016/j.amc.2023.128277
[17] P. Manuel, B. Brešar, and S. Klavžar, The geodesic transversal problem on some networks, Comp. Appl. Math. 42 (2023), no. 1, 59. https://doi.org/10.1007/s40314-023-02199-9
[18] P. Manuel and S. Klavžar, A general position problem in graph theory, Bulletin of the Australian Mathematical Society 98 (2018), no. 2, 177–187. https://doi.org/10.1017/S0004972718000473
19] D. Meierling, Rautenbach D. Protti, F., and A.R. de Almeida, Cycles in complementary prisms, Discrete Appl. Math. 193 (2015), 180–186. https://doi.org/10.1016/j.dam.2015.04.016
[20] R.E. Miller and J.W. Thatcher (eds.), Complexity of computer computations, The IBM Research Symposia Series, Yorktown Heights, N.Y., Plenum Press, 1972.
[21] P.K. Neethu and U. Chandran S V, A note on the convexity number of the complementary prisms of trees, Discrete Appl. Math. 319 (2022), 480–486. https://doi.org/10.1016/j.dam.2021.07.033
[22] P.K. Neethu, U. Chandran S V, M. Changat, and S. Klavˇzar, On the general position number of complementary prisms, Fund. Inform. 178 (2021), no. 3, 267–281. https://doi.org/10.3233/FI-2021-2006
[23] P.K. Neethu, U. Chandran S V, and J.R. Nascimento, On the monophonic convexity in complementary prisms, Discrete Appl. Math. 343 (2024), 224–230. https://doi.org/10.1016/j.dam.2023.10.024
[24] I. Peterin and G. Semanišin, Geodesic transversal problem for join and lexicographic product of graphs, Comp. Appl. Math. 41 (2022), no. 4, 128. https://doi.org/10.1007/s40314-022-01834-1
[25] I. Peterin and S. Semanišin, On maximal shortest paths cover, Mathematics 9 (2021), no. 14, 1592. https://doi.org/10.3390/math9141592 [26] R. Rajeshkumar and A.M. Anto, Fuzzy detour convexity and fuzzy detour covering in fuzzy graphs, Turkish Journal of Computer and Mathematics Education 12 (2021), no. 2, 2170–2175. https://doi.org/10.17762/turcomat.v12i2.1898
[27] , Some domination parameters in intuitionistic fuzzy graphs, American Institute of Physics Conference Series, vol. 2516, 2022, p. 200016. https://doi.org/10.1063/5.0108668
[28] R. Rajeshkumar, A.M. Anto, and V.M.M. Rose, Fuzzy geodetic and detour spectra: Geodetic-Laplacian energy in fuzzy graphs, Eur. J. Pure Appl. Math. 18 (2025), no. 4, 6307. https://doi.org/10.29020/nybg.ejpam.v18i4.6307
[29] S. Rout, P.K. Mishra, and G.K. Das, Total Roman domination and total domination in unit disk graphs, Commun. Comb. Optim. 10 (2025), no. 4, 803–823. https://doi.org/10.22049/cco.2024.28647.1650
[30] D. Roy, S. Klavˇzar, and A. Lakshmanan, Mutual-visibility and general position in double graphs and in Mycielskians, Appl. Math. Comput. 488 (2025), 129131. https://doi.org/10.1016/j.amc.2024.129131
[31] P.J. Slater, Leaves of trees, Congressus Numerantium 14 (1975), 549–559.
[32] E.J. Thomas, U. Chandran S V, J. Tuite, and G. Di Stefano, On monophonic position sets in graphs, Discrete Appli. Math. 354 (2024), 72–82. https://doi.org/10.1016/j.dam.2023.02.021
[33] E.J. Thomas, U. Chandran S V, J. Tuite, and G. Di Stefano, On the general position number of Mycielskian graphs, Discrete Appl. Math. 353 (2024), 29–43. https://doi.org/10.1016/j.dam.2024.03.015
[34] J. Tian and S. Klavžar, Variety of general position problems in graphs, Bull. Malays. Math. Sci. Soc. 48 (2025), no. 1, 5. https://doi.org/10.1007/s40840-024-01788-z [35] 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
[36] J. Tian, K. Xu, and S. Klavžar, The general position number of the cartesian product of two trees, Bull. Aust. Math. Soc. 104 (2021), no. 1, 1–10. https://doi.org/10.1017/S0004972720001276
[37] J. Tuite, E.J. Thomas, and U. Chandran S V, On some extremal position problems for graphs, Ars Math. Contemp. 25 (2025), no. 1, #P1.09. https://doi.org/10.26493/1855-3974.3094.bc6
[38] E.K. Welton, S. Khudairi, and J. Tuite, Lower general position in cartesian products, Commun. Comb. Optim. 10 (2025), no. 1, 110–125. https://doi.org/10.22049/cco.2024.29171.1879
[39] L.M. Zatesko, R. Carmo, A.L. Guedes, A. Zorzi, R.C. Machado, and C.M. Figueiredo, On the chromatic index of complementary prisms, Acta Math. Univ. Comenian. 88 (2019), no. 3, 1071–1077. | ||
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