Mixed metric dimension of graphs

Kelenc, Aleksander; Kuziak, Dorota; Taranenko, Andrej; Yero, Ismael G.

doi:10.18690/978-961-286-113-1.7

Cited by 33 publications

(66 citation statements)

References 17 publications

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“…The mixed metric dimension is the least cardinality of a mixed metric generator; the notion used here to represent is dim m (G). This idea is put forward in [9], and recently it is investigated in [18], where the authors studied some rotationally symmetric graphs, and the mixed metric dimension of P (n, 2) is studied in [19]. The author in [4] computed mixed metric dimension of flower snarks J n , and wheel graphs W n .…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

On Mixed Metric Dimension of Some Path Related Graphs

Raza

2020

IEEE Access

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A vertex k ∈ V G determined two elements (vertices or edges) , m ∈ V G ∪ E G , if d G (k,) = d G (k, m). A set R m of vertices in a graph G is a mixed metric generator for G, if two distinct elements (vertices or edges) are determined by some vertex set of R m. The least number of elements in the vertex set of R m is known as mixed metric dimension, and denoted as dim m (G). In this paper, the mixed metric dimension of some path related graphs is obtained. Those path related graphs are P 2 n the square of a path, T (P n) total graph of a path, the middle graph of a path M (P n), and splitting graph of a path S(P n). We proved that these families of graphs have constant and unbounded mixed metric dimension, respectively. We further presented an improved result for the metric dimension of the splitting graph of a path S(P n). INDEX TERMS Mixed metric dimension, Metric dimension, Edge metric dimension, Path related graphs.

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

On Mixed Metric Dimension of Some Path Related Graphs

Raza

2020

IEEE Access

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“…For more results in this subject or related subjects see [6], [8] and [10]. In this paper, we study the metric dimension of Cayley graphs on dihedral groups and we characterize all of Cayley graphs on dihedral groups whose metric dimension is two.…”

Section: Theorem 13 [4]mentioning

confidence: 99%

On two-dimensional Cayley graphs

Behtoei

Pour

2017

asat

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A subset W of the vertices of a graph G is a resolving set for G when for each (v, w). The cardinality of a minimum resolving set for G is the metric dimension of G. This concept has applications in many diverse areas including network discovery, robot navigation, image processing, combinatorial search and optimization. The problem of finding metric dimension is NP-complete for general graphs but the metric dimension of trees can be obtained using a polynomial time algorithm. In this paper, we investigate the metric dimension of Cayley graphs on dihedral groups and we characterize a family of them. x is N (x) = {y : x ∼ y}. A walk consists of an alternating sequence of vertices and edges http://dx.doi.org/10.29252/asta.4.1.43 MSC(2010): Primary: 05C25. Secondary: 05C75, 20B10.

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“…In this sense, in the way of correctly recognize the edges of a graph, a new parameter was recently introduced in [8]. Another variant on such direction was also presented in [7] where not only edges are recognized between them, but also there is a recognition scheme between any two elements (vertices or edges) of a graph. In this work we only center our attention into recognizing edges.…”

Section: Introductionmentioning

confidence: 99%

Edge Metric Dimension of Some Graph Operations

Peterin

Yero

2019

Bull. Malays. Math. Sci. Soc.

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Let G = (V, E) be a connected graph. Given a vertex v ∈ V and an edge e = uw ∈ E, the distance between v and e is defined as d G (e, v) = min{d G (u, v), d G (w, v)}. A nonempty set S ⊂ V is an edge metric generator for G if for any two edges e 1 , e 2 ∈ E there is a vertex w ∈ S such that d G (w, e 1 ) = d G (w, e 2 ). The minimum cardinality of any edge metric generator for a graph G is the edge metric dimension of G. The edge metric dimension of the join, lexicographic and corona product of graphs is studied in this article.

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Mixed metric dimension of graphs

Cited by 33 publications

References 17 publications

On Mixed Metric Dimension of Some Path Related Graphs

On Mixed Metric Dimension of Some Path Related Graphs

On two-dimensional Cayley graphs

Edge Metric Dimension of Some Graph Operations

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