Variety of mutual-visibility problems in graphs

Cicerone, Serafino (author), Di Stefano, Gabriele (author), Drožđek, Lara (author), Hedžet, Jaka (author), Klavžar, Sandi (author), Yero, Ismael G. (author)

If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)\cap X \subseteq \{u,v\}$. If each two vertices from $X$ are $X$-visible, then $X$ is a mutual-visibility set. The mutual-visibility number of $G$ is the cardinality of a largest mutual-visibility set of $G$ and has been already investigated. In this paper a variety of mutual-visibility problems is introduced based on which natural pairs of vertices are required to be $X$-visible. This yields the total, the dual, and the outer mutual-visibility numbers. We first show that these graph invariants are related to each other and to the classical mutual-visibility number, and then we prove that the three newly introduced mutual-visibility problems are computationally difficult. According to this result, we compute or bound their values for several graphs classes that include for instance grid graphs and tori. We conclude the study by presenting some inter-comparison between the values of such parameters, which is based on the computations we made for some specific families.

• Language: English
• Keywords: mutual-visibility, total mutual-visibility, dual mutual-visibility number, outer mutual-visibility, grid graphs, torus graphs, computational complexity
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.09.2023
• Year: 2023
• Number of pages: 13 str.
• Numbering: Vol. 974, art. no. ǂ114096
• PID: 20.500.12556/RUL-155652
• UDC: 519.17
• ISSN on article: 0304-3975
• DOI: 10.1016/j.tcs.2023.114096
• COBISS.SI-ID: 161787907
• Publication date in RUL: 10.04.2024

Record is a part of a journal

• Title: Theoretical computer science
• Shortened title: Theor. comp. sci.
• Publisher: Elsevier
• ISSN: 0304-3975
• COBISS.SI-ID: 26525952

Licences

• License: CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
• Link: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Description: The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.

Secondary language

• Language: Slovenian
• Keywords: vzajemna vidnost, celotna vzajemna vidnost, število dualne vzajemne vidnosti, število zunanje vzajemne vidnosti, rešetke, torusni grafi, računska zahtevnost