论文标题
笛卡尔产品中总共可见性数量零且总相互可视性的图表
Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products
论文作者
论文摘要
如果$ g $是图形和$ x \ subseteq v（g）$，则$ x $是总共相互可视性集，如果每对顶点$ x $ x $和$ y $ of $ g $ of最短$ x，y $ - $ -path $ p $，$ v（p）\ cap x \ cap x \ subseteq \ subseteq \ subseteq \ {x，x，y \ {x，y \ { $ g $的最大共同可见性集的基数是$ g $的总互及可见度数量$μ_ {\ rm t}（g）$。具有$μ_ {\ rm t}（g）= 0 $的图表的图表为图形，其中没有顶点是convex $ p_3 $的中心顶点。笛卡尔产品的总相互视野数量是有限的，并证明了几个确切的结果。例如，$μ_ {\ rm t}（k_n \，\ square \，k_m）= \ max \ {n，m \} $和$μ_ {\ rm t}（t \，\ square \ square \，h）一棵树和$ h $一个任意图。还证明了$μ_ {\ rm t}（g \，\ square \，h）$可以任意大于$μ_ {\ rm t}（g）μ_ {\ rm t}（h）$。
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 $μ_{\rm t}(G)$ of $G$. Graphs with $μ_{\rm t}(G) = 0$ 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, $μ_{\rm t}(K_n\,\square\, K_m) = \max\{n,m\}$ and $μ_{\rm t}(T\,\square\, H) = μ_{\rm t}(T)μ_{\rm t}(H)$, where $T$ is a tree and $H$ an arbitrary graph. It is also demonstrated that $μ_{\rm t}(G\,\square\, H)$ can be arbitrary larger than $μ_{\rm t}(G)μ_{\rm t}(H)$.
