Local metric dimension of graphs: Generalized hierarchical products and some applications

Abstract

Let G be a graph and S ⊆ V (G). If every two adjacent vertices of G have different metric S-representations, then S is a local metric generator for G. A local metric generator of smallest order is a local metric basis for G, its order is the local metric dimension of G. Lower and upper bounds on the local metric dimension of the generalized hierarchical product are proved and demonstrated to be sharp. The results are applied to determine or bound the dimension of several graphs of importance in mathematical chemistry. Using the dimension, a new model for assigning codes to customers in delivery services is proposed.

Metric dimension related parameters in graphs: A survey on combinatorial, computational and applied results

“…We first want to remark an interesting "application" of local metric dimension in delivery services that has recently appeared in [111]. In such work, authors assumed that a given company could need to assign codes to its customers such that the code of a customer will uniquely determine its location.…”

“…In some sense, it seems to be natural that the company would be interested into making the length of the codes as short as possible. In order to design a graph theory model for this problem, in [111], customers are considered as the vertices of a (an edge-weighted) graph G. Vertices u and v are declared to be adjacent in G if one of the following conditions is fulfilled: (i) there is no other customer on the u, v-geodesics; (ii) the first letters of the family names of the customers u and v are the same. Next, let S = {v 1 , .…”

“…Note that the second part of such pair is precisely r(v|S), i.e., the metric representation of v with respect to S . The authors of [111] made a comparison of such model with a related one from [108] (which uses the classical metric dimension), and found out that the local metric dimension model in general behaves much better, and for instance, in cases like bipartite graphs (where the local metric dimension equals 1, and the classical metric dimension could be much larger) the efficiency of the new model is much higher.…”

Some binary products and integer linear programming for computing $k$-metric dimension of graphs

“…Also, we note that if U = V (G), then the generalized hierarchical product of G and H is the standard Cartesian product G H, cf. [21]. Moreover, if |U | = 1, then the generalized hierarchical product G and H is a cluster product G{H}, see [26,27].…”

Edge metric dimensions via hierarchical product and integer linear programming

“…In this section we consider the edge metric dimension of the hierarchical product of graphs and mention in passing that the metric dimension and the fractional metric dimension of these products were studied in [10], and the local metric dimension in [16].…”