20.500.12556/DiRROS-20228 Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products 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 $\mu_{\rm t}(G)\mu_{\rm t}(H)$. mutual-visibility set total mutual-visibility set bypass vertex Cartesian product of graphs trees množica vzajemne vidnosti množica celotne vzajemne vidnosti obhodno vozlišče kartezični produkt grafov drevesa true false true Angleški jezik Slovenski jezik Neznano 2024-08-26 11:32:55 2024-08-26 11:32:55 2024-08-27 03:35:32 0000-00-00 00:00:00 2024 0 0 Spletna objava: 1. 5. 2023; str. 1277–1291 no. 4 Vol. 44 2024 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2024-01-01 519.17 1234-3099 10.7151/dmgt.2496 204706307 DMGT-2496.pdf DMGT-2496.pdf 1 EDFB9CDC75AA5383D4B976DE59F57264 e99523beaa1d76baf084f2678deb4b14a0e258b9930a93a7bcbab2ae24bd916d 5eb437f8-637e-11ef-b487-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=28333 https://www.dmgt.uz.zgora.pl/publish/article.php?doi=2496 1 6b13fef7-637d-11ef-b487-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=28332 Inštitut za matematiko, fiziko in mehaniko 0 0 0