Mixed metric dimension of some graphs

Abstract

A vertex gv in a connected graph G is said to distinguish two distinct elements p,q∈V(G)⋃E(G) if dG(p,gv)≠dG(q,gv). A subset W⊆V(G) is a mixed metric generator of G if every two distinct elements from V(G)⋃E(G) are distinguished by W. The mixed metric dimension of G, denoted by βm(G), is the minimum cardinality of mixed metric generators in it. In this work, we first answer the problem of characterizing graphs G which achieve βm(G)=n−g(G)+3 in [Appl. Math. Comput. 314 (2017) 429–438] where g(G) is the girth of G, and then determine bounds on the mixed metric dimension of Cartesian product G□H which resolves two open problems in [Discrete Math. 341 (2018) 2083–2088] as a natural corollary. In addition, we provide two closed formulae for βm(T□Pn) in terms of βm(T) which generalize the result on βm(Ps□Pn) in [Appl. Math. Comput. 314 (2017) 429–438].