Resolving dominating partitions in graphs

Abstract

A partition Π = {S 1 , . . . , S k } of the vertex set of a connected graph G is called a resolving partition of G if for every pair of vertices u and v, d(u, S j ) = d(v, S j ), for some part S j . The partition dimension β p (G) is the minimum cardinality of a resolving partition of G. A resolving partition Π is called resolving dominating if for every vertex v of G, d(v, S j ) = 1, for some part S Many variations of location in graphs have since been defined (see survey [23]). For example, in 2000, Chartrand, Salehi and Zhang study the resolvability of graphs in terms of partitions [6], as a generalization of resolving sets when the vertices are classified in different types. A few years later, resolving dominating sets were introduced by Brigham, Chartrand, Dutton and Zhang [1] and independently by Henning and Oellermann [17] as metric-locating-dominating sets, combining the usefulness of resolving sets and dominating sets. Resolving dominating sets have been further studied in [3,11,19]. In this paper, following the ideas of these works, we introduce the resolving dominating partitions, as a way for distinguishing the vertices of a graph by using on the one hand partitions, and on the other hand, both domination and location.

Metric-locating-dominating sets of graphs for constructing related subsets of vertices

“…For example, resolving sets serve as a tool for combinatorial optimization [39], game theory [20], and pharmaceutical chemistry [9]; and dominating sets are helpful to analyze computer networks [38], design codes [12], and model biological networks [23]. Although metric-locating-dominating sets are hard to handle, for entailing the complexity of the other two concepts, they have been studied in several papers, for instance [5,26,27], and further generalized in other works such as [35,44].…”

Neighbor-locating colorings in graphs

“…The partition Π is called a metric-locating-dominating partition, an MLD-partition for short, if it is both dominating and metric-locating. The partition metric-location-domination number η p (G) of G is the minimum cardinality of an MLD-partition of G. In [28], it was proved that β p (G) ≤ η p (G) ≤ β p (G) + 1.…”

“…We examine the neighbor-locating chromatic number for two graph operations: join and disjoint union, and also for two graph families: split graphs and Mycielski graphs.Henning and O. R. Oellermann introduced the so-called metric-locating-dominating sets, by merging the concepts of metric-locating set and dominating set.In [14], G. Chartrand, E. Salehi and P. Zhang, brought the notion of metric location to the ambit of vertex partitions, introducing the resolving partitions, also called metriclocating partition, and defining the partition dimension. Metric location and domination, in the context of vertex partitions, are studied in [28]. In [11], there were introduced the so-called locating colorings considering resolving partitions formed by independents sets.Neighbor location in sets was introduced by P. Slater in [39].…”

“…In [14], G. Chartrand, E. Salehi and P. Zhang, brought the notion of metric location to the ambit of vertex partitions, introducing the resolving partitions, also called metriclocating partition, and defining the partition dimension. Metric location and domination, in the context of vertex partitions, are studied in [28]. In [11], there were introduced the so-called locating colorings considering resolving partitions formed by independents sets.…”

Neighbor-locating coloring: graph operations and extremal cardinalities

“…In [6], G. Chartrand, E. Salehi and P. Zhang, brought the concept of metric-location to the ambit of vertex partitions. Metric-location and domination, in the context of vertex partitions, is studied in [7]. In [3], there were introduced the so-called locating colorings considering partitions formed by independents sets.…”