The doubly metric dimension of some convex polytopes

For a connected graph [Formula: see text], two vertices [Formula: see text] are doubly resolved by [Formula: see text] if [Formula: see text]. The doubly metric dimension of a connected graph [Formula: see text] is defined as the cardinality of a minimum set of vertices that doubly resolves every pair of vertices in [Formula: see text]. It has been shown that the doubly metric dimension problem is NP-hard. In this paper, we study the doubly metric dimensions of some 3-connected plane graphs: the convex polytopes [Formula: see text], [Formula: see text] and [Formula: see text], and we also determine the doubly metric dimensions of their related plane graphs [Formula: see text], [Formula: see text] and [Formula: see text].