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Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products
ID Tian, Jing (Author), ID Klavžar, Sandi (Author)

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Abstract
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 This link opens in a new window
UDC:519.17
ISSN on article:1234-3099
DOI:10.7151/dmgt.2496 This link opens in a new window
COBISS.SI-ID:204706307 This link opens in a new window
Publication date in RUL:26.08.2024
Views:174
Downloads:24
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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 This link opens in a new window

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

Projects

Funder:ARRS - Slovenian Research Agency
Project number:P1-0297
Name:Teorija grafov

Funder:ARRS - Slovenian Research Agency
Project number:J1-2452
Name:Strukturni, optimizacijski in algoritmični problemi v geometrijskih in topoloških predstavitvah grafov

Funder:ARRS - Slovenian Research Agency
Project number:N1-0285
Name:Metrični problemi v grafih in hipergrafih

Funder:Other - Other funder or multiple funders
Funding programme:Postgraduate Research Practice Innovation Program of Jiangsu Province
Project number:KYCX22 0323

Funder:Other - Other funder or multiple funders
Funding programme:Interdisciplinary Innovation Fund for Doctoral Students of Nanjing University of Aeronautics and Astronautics
Project number:KXKCXJJ202204

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