20.500.12556/RUL-160334 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 Članek v reviji 2024-08-26 11:19:05 2024-08-26 11:19:08 2024-08-27 03:52:26 0000-00-00 00:00:00 2024 0 0 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 E4C2E417FFC735678289385C5BB090B7 14de3e347c23eacc8f49e57b4a97ad1bd447c111cdd28b3dca43bf9542318162 f29ae1ba-638e-11ef-8f74-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=188929 https://www.dmgt.uz.zgora.pl/publish/article.php?doi=2496 1 077e19b5-638c-11ef-8f74-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=188925 Fakulteta za matematiko in fiziko 0 0 0