Distance-based vertex identification in graphs: The outer multiset dimension

Gil-Pons, Reynaldo; Ramírez-Cruz, Yunior; Trujillo-Rasúa, Rolando; Yero, Ismael G.

doi:10.1016/j.amc.2019.124612

Cited by 8 publications

(11 citation statements)

References 9 publications

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“…Various distance-related parameters have been studied, such as the partition distance of graphs were studied in [7], Alhevaz et al gave the sharp bounds for the generalized distance spectral radius of graphs [8], Wang studied distance bounds for generalized bicycle codes and Pryadko [9], Nadeem et al found the fault-tolerant partition dimension of oxide interconnection networks [10]. Concerning metric dimensions, that have been of more interest to the research community, one could remark of a few of them (although possibly not all of the most remarkable ones): partition dimension [11], strong metric dimension [12], k-metric dimension [13], identifying codes [14], k-metric anti-dimension [15], local metric dimension [16], edge metric dimension [17] and multiset dimension [18] (see also [19] for the outer-multiset dimension). Each of these variations of the metric dimension mentioned above have been recently studied to a greater or lesser extent, and even some combinations between them have also appeared, including, for instance, k-partition dimension [20], or local edge dimension [21].…”

Section: Literature Reviewmentioning

confidence: 99%

A New Technique to Uniquely Identify the Edges of a Graph

et al. 2023

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Graphs are useful for analysing the structure models in computer science, operations research, and sociology. The word metric dimension is the basis of the distance function, which has a symmetric property. Moreover, finding the resolving set of a graph is NP-complete, and the possibilities of finding the resolving set are reduced due to the symmetric behaviour of the graph. In this paper, we introduce the idea of the edge-multiset dimension of graphs. A representation of an edge is defined as the multiset of distances between it and the vertices of a set, B⊆V(Γ). If the representation of two different edges is unequal, then B is an edge-multiset resolving a set of Γ. The least possible cardinality of the edge-multiset resolving a set is referred to as the edge-multiset dimension of Γ. This article presents preliminary results, special conditions, and bounds on the edge-multiset dimension of certain graphs. This research provides new insights into structure models in computer science, operations research, and sociology. They could have implications for developing computer algorithms, aircraft scheduling, and species movement between regions.

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Section: Literature Reviewmentioning

confidence: 99%

A New Technique to Uniquely Identify the Edges of a Graph

et al. 2023

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“…In order to not deal with such situation, and have a parameter that can be computed for every graph, in [78] it was defined a related parameter as follows. The set S ⊂ V (G) is an outer multiset resolving set if for any two distinct vertices u, v ∈ V (G) \ S it follows that m(u|S) = m(v|S).…”

Section: Multiset Versions Of Metric Dimensionmentioning

confidence: 99%

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

Kuziak,

Yero

2021

Preprint

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“…While applications have been continuously appearing, this invariant has also been theoretically studied in a high number of other papers, which we do not mention here due to the high amount of them. Moreover, several variations of metric generators including resolving dominating sets [10], independent resolving sets [11], local metric generators [12], strong resolving sets [3], k-metric generators [13], edge metric generators [14], mixed metric generators [15], antiresolving sets [16], multiset resolving sets [17], metric colorings [18], resolving partitions [19] and strong resolving partitions [20] have been introduced and studied. The last three cases are remarkable in the sense that they concern with a partition of the vertex set of the graph which uniquely identifies every vertex of the graph.…”

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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Distance-based vertex identification in graphs: The outer multiset dimension

Cited by 8 publications

References 9 publications

A New Technique to Uniquely Identify the Edges of a Graph

A New Technique to Uniquely Identify the Edges of a Graph

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

Further new results on strong resolving partitions for graphs

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