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517. 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: 228; Downloads: 182
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519. 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: 211; Downloads: 101
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520. Persistent homology with selective Rips complexes detects geodesic circlesŽiga Virk, 2024, original scientific article Abstract: This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of $S^1$) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space using persistent homology. Under fairly mild conditions, we show that such a loop either terminates a 1- dimensional homology class or gives rise to a 2-dimensional homology class in persistent homology. The main tool in this detection technique are selective Rips complexes, new custom made complexes that function as an appropriate combinatorial lens for persistent homology to detect the above mentioned loops. The main argument is based on a new concept of a local winding number, which turns out to be an invariant of certain homology classes.
Keywords: simple closed geodesic, Rips complex, persistent homology, local winding number
Published in DiRROS: 26.08.2024; Views: 220; Downloads: 93
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