Strong resolving partitions for strong product graphs and Cartesian product graphs

Yero, Ismael G.

doi:10.1016/j.dam.2015.08.026

Cited by 3 publications

(1 citation statement)

References 25 publications

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“…A strong resolving partition of minimum cardinality is called a strong partition basis, and its cardinality the strong partition dimension of G, denoted by pd s (G). The strong partition dimension of graphs was introduced in [20] and further studied in [22].…”

Section: Introductionmentioning

confidence: 99%

Further new results on strong resolving partitions for graphs

Kuziak

Yero

2020

Open Mathematics

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Abstract A set W of vertices of a connected graph G strongly resolves two different vertices x, y ∉ W if either d G (x, W) = d G (x, y) + d G (y, W) or d G (y, W) = d G (y, x) + d G (x, W), where d G (x, W) = min{d(x,w): w ∈ W} and d(x,w) represents the length of a shortest x − w path. An ordered vertex partition Π = {U 1, U 2,…,U k } of a graph G is a strong resolving partition for G, if every two different vertices of G belonging to the same set of the partition are strongly resolved by some other set of Π. The minimum cardinality of any strong resolving partition for G is the strong partition dimension of G. In this article, we obtain several bounds and closed formulae for the strong partition dimension of some families of graphs and give some realization results relating the strong partition dimension, the strong metric dimension and the order of graphs.

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Section: Introductionmentioning

confidence: 99%

Further new results on strong resolving partitions for graphs

Kuziak

Yero

2020

Open Mathematics

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Strong resolving graph of a zero-divisor graph

Nikandish

Nikmehr

Bakhtyiari

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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On the k-partition dimension of graphs

Estrada‐Moreno

2020

Theoretical Computer Science

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As a generalization of the concept of the partition dimension of a graph, this article introduces the notion of the k-partition dimension. Given a nontrivial connected graph G = (V, E), a partition Π of V is said to be a k-partition generator of G if any pair of different vertices u, v ∈ V is distinguished by at least k vertex sets of Π, i.e., there exist at least k vertex sets S 1 , . . . , S k ∈ Π such that d(u, S i ) = d(v, S i ) for every i ∈ {1, . . . , k}. A k-partition generator of G with minimum cardinality among all their k-partition generators is called a k-partition basis of G and its cardinality the k-partition dimension of G. A nontrivial connected graph G is k-partition dimensional if k is the largest integer such that G has a k-partition basis. We give a necessary and sufficient condition for a graph to be r-partition dimensional and we obtain several results on the k-partition dimension for k ∈ {1, . . . , r}.

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Strong resolving partitions for strong product graphs and Cartesian product graphs

Cited by 3 publications

References 25 publications

Further new results on strong resolving partitions for graphs

Further new results on strong resolving partitions for graphs

Strong resolving graph of a zero-divisor graph

On the k-partition dimension of graphs

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