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In this paper a new graph invariant based on the minimal hitting set problemis introduced. It is shown that it represents a tight lower bound for thedoubly metric dimension of a graph. Exact values of new invariant for paths,stars, complete graphs and complete bipartite graph are obtained. The paperanalyzes some tight bounds for the new invariant in general case. Also severalextremal differences between some related invariants are determined.

A new lower bound for doubly metric dimension and related extremal differences