Distance mutual-visibility coloring: relations with (total) domination, exact distance graphs and graph products

Abstract

The concept of mutual-visibility (MV) has been extended in several directions. A vertex subset $S$ of a graph $G$ is a $k$-distance mutual-visibility ($k$DMV) set if for any two vertices in $S$, there is a geodesic between them of length at most $k$ whose internal vertices are not in $S$. In this paper, we combine this with the MV coloring as follows. For any integer $k\geq1$, a $k$DMV coloring of $G$ is a partition of $V(G)$ into $k$DMV sets, and the $k$DMV chromatic number $\chi_{\mu_k}(G)$ is the minimum cardinality of such a partition. When $k=1$ or $k\ge {\rm diam}(G)$, it equals the clique cover number $\theta(G)$ or the MV chromatic number $\chi_{\mu}(G)$, respectively. So, our attention is given to $1<k<{\rm diam}(G)$ with $k=2$ producing the most interesting results. We prove that $\chi_{\mu_2}(G)\le|V(G)|/2$ and present large families of graphs that attain the bound. In addition, $\chi_{\mu_2}(G)$ is bounded from above by the total domination number $\gamma_t(G)$ if $G$ is isolate-free, while in graphs $G$ with girth $g(G)\geq7$, $\chi_{\mu_2}(G)$ is bounded from below by the domination number $\gamma(G)$. A surprising relation with the exact distance-2 graphs is found, which results in $\theta(G^{[\natural2]})=\gamma_{t}(G)$ for any isolate-free graph $G$ with $g(G)\geq7$. The relation is explored further in lexicographic product graphs, where we prove the sharp inequalities $\chi_{\mu_{2}}(G\circ H)\leq \theta(G^{[\natural2]})\leq \theta\big{(}(G\circ H)^{[\natural2]}\big{)}$. We also prove a sharp lower (resp. upper) bound on $\chi_{\mu_2}$ (resp. $\chi_{\mu_k}$) for the Cartesian (resp. strong) product of two connected graphs and show that they are widely sharp. Finally, we characterize the block graphs $G$ with $\chi_{\mu_k}(G)=\chi_{\mu}(G)$, where $k={\rm diam}(G)-1$.