Mutual-visibility problems on graphs of diameter two

Cicerone, Serafino (author), Di Stefano, Gabriele (author), Klavžar, Sandi (author), Yero, Ismael G. (author)

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.

• Language: English
• Keywords: mutual-visibility set, mutual-visibility number, diameter-two graphs, line graphs, cographs
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: 16 str.
• Numbering: Vol. 120, art. 103995
• PID: 20.500.12556/RUL-158163
• UDC: 519.17
• ISSN on article: 0195-6698
• DOI: 10.1016/j.ejc.2024.103995
• COBISS.SI-ID: 196753923
• Publication date in RUL: 27.05.2024

CICERONE, Serafino, DI STEFANO, Gabriele, KLAVŽAR, Sandi and YERO, Ismael G., 2024, Mutual-visibility problems on graphs of diameter two. European journal of combinatorics [online]. 2024. Vol. 120, no. 103995. [Accessed 4 June 2025]. DOI 10.1016/j.ejc.2024.103995. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=158163

Record is a part of a journal

• Title: European journal of combinatorics
• Shortened title: Eur. j. comb.
• Publisher: Elsevier
• ISSN: 0195-6698
• COBISS.SI-ID: 25427968

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: množica vzajemne vidnosti, število vzajemne vidnosti, grafi premera dva, grafi povezav, kografi