Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products

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 ${\mu }_{\mathrm{t}}(G)$ of $G$. Graphs with ${\mu }_{\mathrm{t}}(G)=0$ 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, ${\mu }_{\mathrm{t}}(K_n\square K_m)=\max \{n,m\}$ and ${\mu }_{\mathrm{t}}(T\square H)={\mu }_{\mathrm{t}}(T){\mu }_{\mathrm{t}}(H)$, where $T$ is a tree and $H$ an arbitrary graph. It is also demonstrated that ${\mu }_{\mathrm{t}}(G\square H)$ can be arbitrary larger than ${\mu }_{\mathrm{t}}(G){\mu }_{\mathrm{t}}(H)$.