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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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https://www.dmgt.uz.zgora.pl/publish/article.php?doi=2496
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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
UDC:
519.17
ISSN on article:
1234-3099
DOI:
10.7151/dmgt.2496
COBISS.SI-ID:
204706307
Publication date in RUL:
26.08.2024
Views:
174
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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
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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