20.500.12556/RUL-158163 Mutual-visibility problems on graphs of diameter two The mutual-visibility problem in a graph $G$ asks for the cardinality of a largest set of vertices $S\subseteq V(G)$ so that for any two vertices $x,y \in S$ there is a shortest $x,y$-path $P$ so that all internal vertices of $P$ are not in $S$. This is also said as $x,y$ are visible with respect to $S$, or $S$-visible for short. Variations of this problem are known, based on the extension of the visibility property of vertices that are in and/or outside $S$. Such variations are called total, outer and dual mutual-visibility problems. This work is focused on studying the corresponding four visibility parameters in graphs of diameter two, throughout showing bounds and/or closed formulae for these parameters. The mutual-visibility problem in the Cartesian product of two complete graphs is equivalent to (an instance of) the celebrated Zarankiewicz's problem. Here we study the dual and outer mutual-visibility problem for the Cartesian product of two complete graphs and all the mutual-visibility problems for the direct product of such graphs as well. We also study all the mutual-visibility problems for the line graphs of complete and complete bipartite graphs. As a consequence of this study, we present several relationships between the mentioned problems and some instances of the classical Turán problem. Moreover, we study the visibility problems for cographs and several non-trivial diameter-two graphs of minimum size. mutual-visibility set mutual-visibility number diameter-two graphs line graphs cographs množica vzajemne vidnosti število vzajemne vidnosti grafi premera dva grafi povezav kografi true false true Angleški jezik Slovenski jezik Članek v reviji 2024-05-27 10:26:13 2024-05-27 10:26:17 2024-06-21 03:40:28 0000-00-00 00:00:00 2024 0 0 16 str. art. 103995 Vol. 120 Aug. 2024 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17 0195-6698 10.1016/j.ejc.2024.103995 196753923 1-s2.0-S0195669824000805-main.pdf 1-s2.0-S0195669824000805-main.pdf 1 B90F072AECA8B74805D2A3ECE722E33B c051d6ac805fb14e712f14856b2618c03a765574bcb38c8a29f59e6fdeb51f6e 97067124-1c04-11ef-927e-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=185636 https://www.sciencedirect.com/science/article/pii/S0195669824000805 1 c9ea7224-1c02-11ef-927e-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=185635 Fakulteta za matematiko in fiziko 0 0 0