The metric dimension of circulant graphs and their Cartesian products

Chau, Kevin; Gosselin, Shonda

doi:10.7494/opmath.2017.37.4.509

Cited by 16 publications

(22 citation statements)

References 7 publications

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“…. , v 11 Let W ′ � u 1 , u 4 , u 7 . en, the representations of the vertices v i : 12 ≤ i ≤ n of G n [1,4] with respect to W ′ and the vertex u 10 are calculated as follows:…”

Section: If Two Vertices From V I and Onementioning

confidence: 99%

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Resolvability in Subdivision of Circulant Networks Cn1,k

Wei

Bokhary

Abbas

et al. 2020

Discrete Dynamics in Nature and Society

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Circulant networks form a very important and widely explored class of graphs due to their interesting and wide-range applications in networking, facility location problems, and their symmetric properties. A resolving set is a subset of vertices of a connected graph such that each vertex of the graph is determined uniquely by its distances to that set. A resolving set of the graph that has the minimum cardinality is called the basis of the graph, and the number of elements in the basis is called the metric dimension of the graph. In this paper, the metric dimension is computed for the graph Gn1,k constructed from the circulant graph Cn1,k by subdividing its edges. We have shown that, for k=2, Gn1,k has an unbounded metric dimension, and for k=3 and 4, Gn1,k has a bounded metric dimension.

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“…. , v 11 Let W ′ � u 1 , u 4 , u 7 . en, the representations of the vertices v i : 12 ≤ i ≤ n of G n [1,4] with respect to W ′ and the vertex u 10 are calculated as follows:…”

Section: If Two Vertices From V I and Onementioning

confidence: 99%

“…e families of graphs having constant metric dimension are studied in [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Resolvability in Subdivision of Circulant Networks Cn1,k

Wei

Bokhary

Abbas

et al. 2020

Discrete Dynamics in Nature and Society

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“…M. Ali found the metric dimension of two families of a graph with constant metric dimension [13], Maryum calculated the metric dimension of path related graphs [14], Shahida proved metric dimension of the join of two graphs [15]. Similarly a lot of work is done by [12], [13], [16], [17], [18], [19], [23], [24].…”

Section: Introductionmentioning

confidence: 99%

Computing Metric Dimension of Power of Total Graph

Nawaz

Ali²,

Khan³

et al. 2021

IEEE Access

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For a connected graph G, the distance d (u, v) between any two vertices u, v ∈ V(G) is defined as; d(u, v) = min Pu,v {length of P uv } i.e the minimum length of path connecting any two vertices. Two vertices u, v of a graph G are said to be resolved by a verticesIf the representation of all the vertices of graph G are different w.r.t W then W is the resolving set for G [1]. A resolving set that contains minimum numbers of vertices is metric basis for G. The cardinality of smallest resolving set is the metric dimension of G, represented by dim(G) [2], [3].A family of connected graph G has unbounded metric dimension if dim(G) depends on order of a graph. In this paper, we show that total graph of cycle power three and four have unbounded metric dimension. We also proved some results on edges of power of cycle graph.

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“…In the second section, we will compute the edge metric dimension of the family of circulant graphs C n (1,2). In third section, we will calculate the edge metric dimension of the family of circulant graphs C n (1,3). In last section, we concluded the paper by open problem.…”

Section: Introductionmentioning

confidence: 99%

“…3 Edge metric dimension of family of circulant graphs C n (1,3) In this section, we will compute the edge metric dimension of family of circulant graphs C n (1, 3). Figure 2 shows the circulant graph C 16 (1,3).…”

Section: Introductionmentioning

confidence: 99%

Edge metric dimension of some classes of circulant graphs

Ahsan

Zahid

Zafar

2020

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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Let G = (V (G), E(G)) be a connected graph and x, y ∈ V (G), d(x, y) = min{ length of x − y path } and for e ∈ E(G), d(x, e) = min{d(x, a), d(x, b)}, where e = ab. A vertex x distinguishes two edges e1 and e2, if d(e1, x) ≠ d(e2, x). Let WE = {w1, w2, . . ., wk} be an ordered set in V (G) and let e ∈ E(G). The representation r(e | WE) of e with respect to WE is the k-tuple (d(e, w1), d(e, w2), . . ., d(e, wk)). If distinct edges of G have distinct representation with respect to WE, then WE is called an edge metric generator for G. An edge metric generator of minimum cardinality is an edge metric basis for G, and its cardinality is called edge metric dimension of G, denoted by edim(G). The circulant graph Cn(1, m) has vertex set {v1, v2, . . ., vn} and edge set {vivi+1 : 1 ≤ i ≤ n−1}∪{vnv1}∪{vivi+m : 1 ≤ i ≤ n−m}∪{vn−m+ivi : 1 ≤ i ≤ m}. In this paper, it is shown that the edge metric dimension of circulant graphs Cn(1, 2) and Cn(1, 3) is constant.

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The metric dimension of circulant graphs and their Cartesian products

Cited by 16 publications

References 7 publications

Resolvability in Subdivision of Circulant Networks Cn1,k

Resolvability in Subdivision of Circulant Networks Cn1,k

Computing Metric Dimension of Power of Total Graph

Edge metric dimension of some classes of circulant graphs

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