Some new general lower bounds for mixed metric dimension of graphs

Abstract

A vertex $w \in V$ resolves two elements $x,y \in V \cup E$ if $d(w,x) \neq d(w,y)$. The mixed resolving set is a set of vertices $S$, $S\subseteq V$ if any two elements of $E \cup V$ are resolved by some element of $S$. A minimal resolving set related to inclusion is called mixed resolving basis, and its cardinality is called the mixed metric dimension of a graph $G$. This paper introduces three new general lower bounds for the mixed metric dimension of a graph. The exact values of mixed metric dimension for torus graph are determined using one of these lower bounds. Finally, some illustrative examples of these new lower bounds and those known in the literature are presented on a set of some well-known graphs.