Mixed metric dimension of graphs

TL;DR

This paper introduces the concept of mixed metric dimension in graphs, characterizes graphs with extremal values, explores its properties in various graph families, and proves the NP-hardness of computing it.

Contribution

It defines mixed metric dimension, characterizes graphs with extremal values, and establishes NP-hardness of its computation.

Findings

Characterization of graphs with extremal mixed metric dimension

Upper bounds related to graph girth

NP-hardness of determining mixed metric dimension

Abstract

Let $G=(V,E)$ be a connected graph. A vertex $w\in V$ distinguishes two elements (vertices or edges) $x,y\in E\cup V$ if $d_G(w,x)\ne d_G(w,y)$. A set $S$ of vertices in a connected graph $G$ is a mixed metric generator for $G$ if every two elements (vertices or edges) of $G$ are distinguished by some vertex of $S$. The smallest cardinality of a mixed metric generator for $G$ is called the mixed metric dimension and is denoted by $\mathrm{mdim}(G)$. In this paper we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds. We also give results about the mixed metric dimension of some families of graphs and present an upper bound with respect to the girth of a graph. Finally, we prove that the problem of determining the mixed metric dimension of a graph is NP-hard in the general case.