| Title: | Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products |
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| Authors: | ID Tian, Jing (Author)
ID Klavžar, Sandi (Author) |
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| Files: |  PDF - Presentation file, download (184,44 KB)
MD5: EDFB9CDC75AA5383D4B976DE59F57264
 URL - Source URL, visit https://www.dmgt.uz.zgora.pl/publish/article.php?doi=2496
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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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 $\mu_{\rm t}(G)\mu_{\rm t}(H)$. |
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| Keywords: | mutual-visibility set, total mutual-visibility set, bypass vertex, Cartesian product of graphs, trees |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2024 |
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| Year of publishing: | 2024 |
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| Number of pages: | str. 1277–1291 |
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| Numbering: | Vol. 44, no. 4 |
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| PID: | 20.500.12556/DiRROS-20228  |
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| UDC: | 519.17 |
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| ISSN on article: | 1234-3099 |
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| DOI: | 10.7151/dmgt.2496 |
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| COBISS.SI-ID: | 204706307 |
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| Note: | Spletna objava: 1. 5. 2023;
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| Publication date in DiRROS: | 26.08.2024 |
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| Views: | 243 |
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| Downloads: | 117 |
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