Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products

Tian, Jing (author), Klavžar, Sandi (author)

If $G$ is a graph and $X\subseteq V(G)$, 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 $V(P) \cap X \subseteq \{x,y\}$. The cardinality of a largest total mutual-visibility set of $G$ is the total mutual-visibility number $\mu_{\rm t}(G)$ of $G$. Graphs with $\mu_{\rm t}(G) = 0$ are characterized as the graphs in which no 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, $\mu_{\rm t}(K_n\,\square\, K_m) = \max\{n,m\}$ and $\mu_{\rm t}(T\,\square\, H) = \mu_{\rm t}(T)\mu_{\rm t}(H)$, where $T$ is a tree and $H$ an arbitrary graph. It is also demonstrated that $\mu_{\rm t}(G\,\square\, H)$ can be arbitrary larger than $\mu_{\rm t}(G)\mu_{\rm t}(H)$.

• Language: English
• Keywords: mutual-visibility set, total mutual-visibility set, bypass vertex, Cartesian product of graphs, trees
• 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: 01.01.2024
• Year: 2024
• Number of pages: Str. 1277–1291
• Numbering: Vol. 44, no. 4
• PID: 20.500.12556/RUL-160334
• UDC: 519.17
• ISSN on article: 1234-3099
• DOI: 10.7151/dmgt.2496
• COBISS.SI-ID: 204706307
• Publication date in RUL: 26.08.2024

Record is a part of a journal

• Title: Discussiones mathematicae. Graph theory
• Shortened title: Discuss. Math., Graph Theory
• Publisher: Technical University Press
• ISSN: 1234-3099
• COBISS.SI-ID: 7487065

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: množica vzajemne vidnosti, množica celotne vzajemne vidnosti, obhodno vozlišče, kartezični produkt grafov, drevesa