Mixed metric dimension of graphs

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

doi:10.1016/j.amc.2017.07.027

Cited by 59 publications

(30 citation statements)

References 17 publications

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“…Next, we present some already known results for β m , Proposition 1: [23] For a path graph (P n ) order n ≥ 4, we have β m (P n ) = 2. Proposition 2: [23] Let us consider any two positive integers: e, f β m (K e,f ) = e + f − 1, if e = 2 or f = 2; e + f − 2, otherwise. Proof: For P(n, 1), this holds, and, by the same intuition, this must be true for P(n, 2).…”

Section: Known Resultsmentioning

confidence: 99%

“…The least cardinality of a mixed metric generator for a graph is termed as a mixed metric dimension, denoted as β m (Ŵ). The idea is recently brought forward by Kelenc et al [23].…”

Section: Introductionmentioning

confidence: 99%

“…The notion of a mixed metric dimension clearly indicates that a mixed metric generator is also a standard metric generator and an edge metric generator, The following relationship is given in [23],…”

Section: Introductionmentioning

confidence: 99%

“…The following remark shows the structure of mixed metric dimension: Remark 1: [23] Suppose for some graph Ŵ we have 2 ≤ β m ≤ n. Recently, this concept has attracted some attention, and it has been studied by Raza et al [24]. The authors discussed the mixed metric dimension among the prism graphs, which are commonly known as generalizes Petersen graphs P(n, 1), and two families of convex polytopes A n , R n , further presenting the problem of finding β m (P(n, 2)).…”

Section: Introductionmentioning

confidence: 99%

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Computing the Mixed Metric Dimension of a Generalized Petersen Graph P(n, 2)

Raza

2020

Front. Phys.

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Let Ŵ = (V, E) be a connected graph. A vertex i ∈ V recognizes two elements (vertices or edges) j, k ∈ E ∩ V, if d Ŵ (i, j) = d Ŵ (i, k). A set S of vertices in a connected graph Ŵ is a mixed metric generator for Ŵ if every two distinct elements (vertices or edges) of Ŵ are recognized by some vertex of S. The smallest cardinality of a mixed metric generator for Ŵ is called the mixed metric dimension and is denoted by β m. In this paper, the mixed metric dimension of a generalized Petersen graph P(n, 2) is calculated. We established that a generalized Petersen graph P(n, 2) has a mixed metric dimension equivalent to 4 for n ≡ 0, 2(mod4), and, for n ≡ 1, 3(mod4), the mixed metric dimension is 5. We thus determine that each graph of the family of a generalized Petersen graph P(n, 2) has a constant mixed metric dimension.

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Section: Known Resultsmentioning

confidence: 99%

“…The least cardinality of a mixed metric generator for a graph is termed as a mixed metric dimension, denoted as β m (Ŵ). The idea is recently brought forward by Kelenc et al [23].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computing the Mixed Metric Dimension of a Generalized Petersen Graph P(n, 2)

Raza

2020

Front. Phys.

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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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Monitoring the Edges of a Graph Using Distances

Foucaud

Klasing

Miller

et al. 2020

Lecture Notes in Computer Science

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We introduce a new graph-theoretic concept in the area of network monitoring. A set M of vertices of a graph G is a distance-edge-monitoring set if for every edge e of G, there is a vertex x of M and a vertex y of G such that e belongs to all shortest paths between x and y. We denote by dem(G) the smallest size of such a set in G. The vertices of M represent distance probes in a network modeled by G; when the edge e fails, the distance from x to y increases, and thus we are able to detect the failure. It turns out that not only we can detect it, but we can even correctly locate the failing edge. In this paper, we initiate the study of this new concept. We show that for a nontrivial connected graph G of order n, 1 ≤ dem(G) ≤ n − 1 with dem(G) = 1 if and only if G is a tree, and dem(G) = n − 1 if and only if it is a complete graph. We compute the exact value of dem for grids, hypercubes, and complete bipartite graphs. Then, we relate dem to other standard graph parameters. We show that dem(G) is lowerbounded by the arboricity of the graph, and upper-bounded by its vertex cover number. It is also upper-bounded by twice its feedback edge set number. Moreover, we characterize connected graphs G with dem(G) = 2. Then, we show that determining dem(G) for an input graph G is an NP-complete problem, even for apex graphs. There exists a polynomial-time logarithmic-factor approximation algorithm, however it is NP-hard to compute an asymptotically better approximation, even for bipartite graphs of small diameter and for bipartite subcubic graphs. For such instances, the problem is also unlikey to be fixed parameter tractable when parameterized by the solution size. * This paper is dedicated to the memory of Mirka Miller, who sadly passed away in January 2016. This research was started when Mirka Miller and Joe Ryan visited Ralf Klasing at the LaBRI, University of Bordeaux, in April 2015. † A preliminary version of this paper appeared as [12]. ‡ The authors acknowledge the financial support from the ANR project HOSIGRA (ANR-17-CE40-0022), the IFCAM project "Applications of graph homomorphisms" (MA/IFCAM/18/39), and the Programme IdEx Bordeaux-SysNum (ANR-10-IDEX-03-02).

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Mixed metric dimension of graphs

Cited by 59 publications

References 17 publications

Computing the Mixed Metric Dimension of a Generalized Petersen Graph P(n, 2)

Computing the Mixed Metric Dimension of a Generalized Petersen Graph P(n, 2)

Further new results on strong resolving partitions for graphs

Monitoring the Edges of a Graph Using Distances

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