New results on metric-locating-dominating sets of graphs

TL;DR

This paper explores the properties and relationships of metric-locating-dominating sets in graphs, providing new bounds and characterizations, especially for trees, and connecting them with other graph sets.

Contribution

It introduces novel characterizations of trees based on metric-locating-dominating sets and develops methods to relate these sets to other important graph sets, yielding new bounds.

Findings

Characterization of trees based on metric-location-domination and other parameters

Methods to transform metric-locating-dominating sets into other sets

New bounds on the sizes of various graph sets

Abstract

A dominating set $S$ of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of $S$, and the minimum cardinality of such a set is called the metric-location-domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominating sets to other special sets: resolving sets, dominating sets, locating-dominating sets and doubly resolving sets. We first characterize classes of trees according to certain relationships between their metric-location-domination number and their metric dimension and domination number. Then, we show different methods to transform metric-locating-dominating sets into locating-dominating sets and doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them involving parameters that have not been related so far.