On the partition dimension of trees

Rodríguez-Velázquez, Juan A.; Yero, Ismael G.; Lemańska, Magdalena

doi:10.1016/j.dam.2013.09.026

Cited by 39 publications

(18 citation statements)

References 15 publications

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“…The partition Π is called a metric-locating partition, an ML-partition for short, if, for any pair of distinct vertices u, v ∈ V (G), r(u|Π) = r(v|Π). The partition dimension β p (G) of G is the minimum cardinality of an ML-partition of G. Metric-locating partitions were introduced in [14], and further studied in several papers: bounds [10], graph families [16,17,20,21,24,29,30,31,36,37,41,42] and graph operations [2,9,15,18,19,35,46,47].…”

Section: Locating Partitionsmentioning

confidence: 99%

Neighbor-locating colorings in graphs

Alcón

Gutiérrez

Hernando

et al. 2020

Theoretical Computer Science

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A k-coloring of a graph G is a k-partition Π = {S 1 , . . . , S k } of V (G) into independent sets, called colors. A k-coloring is called neighbor-locating if for every pair of vertices u, v belonging to the same color S i , the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v. The neighbor-locating chromatic number χ N L (G) is the minimum cardinality of a neighbor-locating coloring of G.We establish some tight bounds for the neighbor-locating chromatic number of a graph, in terms of its order, maximum degree and independence number. We determine all connected graphs of order n ≥ 5 with neighbor-locating chromatic number n or n − 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]. Given a graph G, a set S ⊆ V (G) is a dominating set if every vertex not in S is adjacent to some vertex in S. A set S ⊆ V (G) is a locating-dominating set if S is a dominating set and N (u) ∩ S = N (v) ∩ S for every two different vertices u and v not in S. The location-domination number of G, denoted by λ(G), is the minimum cardinality of a locating-dominating set. In [8,27], bounds for this parameter are given. In this paper, merging the concepts studied in [11,39], we introduce the neighbor-locating colorings and the neighbor-locating chromatic number, and examine this parameter in some families of graphs.

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Section: Locating Partitionsmentioning

confidence: 99%

Neighbor-locating colorings in graphs

Alcón

Gutiérrez

Hernando

et al. 2020

Theoretical Computer Science

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“…The partition dimension β p (G) of G is the minimum cardinality of a resolving partition of G. Resolving partitions were introduced in [6], and further studied in [7,9,12,13,22,28]. In some of these papers the partition dimension of G is denoted by pd(G).…”

Section: Metric Dimension and Partition Dimensionmentioning

confidence: 99%

Resolving dominating partitions in graphs

Hernando

Mora

Pelayo

2019

Discrete Applied Mathematics

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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.

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“…As, the problem of finding the partition dimension is a generalize variant of metric dimension, therefore partition dimension is also a NPhard problem. For the discussion of graphs with partition dimension n − 3 we refer [5], graphs obtained by sum operation of cycle and path graph and its partition dimension studied in [17], [39] provide bounds of partition dimension, [14], [28] discussed circulant graph's partitioning and for complete multipartite graph [36], strong partition dimension discussed in [25], [34], whereas its local version dealt in [1], brief and detailed review of resolving partition and partition dimension, we referred the literature [2], [13], [16], [21], [31]- [33], [35], [40]. The applications of resolving partition can be found in different fields such as robot navigation [24], Djokovic-Winkler relation [8], network discovery and verification [6], in chemistry for representing chemical compounds [22], [23], strategies for the mastermind game [12] and in problems of pattern recognition and image processing, some of which involve the use of hierarchical data structures [30] for more applications see [9], [15].…”

Section: Introductionmentioning

confidence: 99%