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548. Role of isotropic lipid phase in the fusion of photosystem II membranesKinga Böde, Uroš Javornik, Ondřej Dlouhý, Ottó Zsiros, Avratanu Biswas, Ildikó Domonkos, Primož Šket, Václav Karlický, Bettina Ughy, Petar H. Lambrev, Vladimír Špunda, Janez Plavec, Győző Garab, 2024, original scientific article Published in DiRROS: 27.08.2024; Views: 235; Downloads: 183
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550. Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian productsJing Tian, Sandi Klavžar, 2024, original scientific article 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 $\mu_{\rm t}(G)\mu_{\rm t}(H)$.
Keywords: mutual-visibility set, total mutual-visibility set, bypass vertex, Cartesian product of graphs, trees
Published in DiRROS: 26.08.2024; Views: 214; Downloads: 104
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