Some binary products and integer linear programming for computing $k$-metric dimension of graphs

Abstract

Let $G$ be a connected graph. For an ordered set $S=\{v_1,\ldots, v_\ell\}\subseteq V(G)$, the vector $r_G(v|S) = (d_G(v_1,v), \ldots, d_G(v_\ell,v))$ is called the metric $S$-representation of $v$. If for any pair of different vertices $u,v\in V(G)$, the vectors $r(v|S)$ and $r(u|S)$ differ in at least $k$ positions, then $S$ is a $k$-metric generator for $G$. A smallest $k$-metric generator for $G$ is a $k$-{\em metric basis} for $G$, its cardinality being the $k$-metric dimension of $G$. A sharp upper bound and a closed formulae for the $k$-metric dimension of the hierarchical product of graphs is proved. Also, sharp lower bounds for the $k$-metric dimension of the splice and link products of graphs are presented. An integer linear programming model for computing the $k$-metric dimension and a $k$-metric basis of a given graph is proposed. These results are applied to bound or to compute the $k$-metric dimension of some classes of graphs that are of interest in mathematical chemistry.