On the edge dimension of a graph

Zubrilina, Nina

doi:10.1016/j.disc.2018.04.010

Cited by 78 publications

(59 citation statements)

References 7 publications

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“…sharpening the bound of k 2 + kD k−1 + D k from Zubrilina (2018). We also prove that there is no subgraph of diameter D with more than (D + 1) k vertices in any connected graph of metric dimension k, and there is no subgraph of diameter D with more than (D + 1) k edges in any connected graph of edge metric dimension k.…”

mentioning

confidence: 78%

“…In particular, Hernando et al [7] proved that the maximum possible number of vertices in a graph of diameter D and metric dimension k is at most (⌊ 2D [3] proved that graphs with edge metric dimension k and diameter D have at most (D + 1) k edges. Later, Zubrilina sharpened the bound for edge metric dimension to k 2 + kD k−1 + D k [9]. Kelenc et al [3] also bounded the edge metric dimension of several classes of graphs, including 2-dimensional grid graphs and d-dimensional hypercube graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Kelenc et al [3] also bounded the edge metric dimension of several classes of graphs, including 2-dimensional grid graphs and d-dimensional hypercube graphs. They asked for a characterization of all graphs with edge metric dimension n − 1, which Zubrilina found [9]. Using the characterization, Zubrilina proved that such graphs have diameter at most 2.…”

Section: Introductionmentioning

confidence: 99%

“…[9] Let G(V, E) be a graph with |V | = n. Then edim(G) = n − 1 if and only if for any distinct v 1 , v 2 ∈ V there exists u ∈ V such that {v 1 , u} ∈ E, {v 2 , u} ∈ E, and u is adjacent to all non-mutual neighbors of v 1 , v 2 .…”

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confidence: 99%

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Metric dimension and pattern avoidance in graphs

Geneson

2018

Preprint

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confidence: 78%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Metric dimension and pattern avoidance in graphs

Geneson

2018

Preprint

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“…Moreover, they have raised many open problems related to edge metric dimension. Furthermore, in 2019, Zubrilina has classified some graphs on the same topic which has the edim(G) � |V| − 1 in her paper [23]. Recently, in 2019, Rafiullah et al studied the edge metric dimensions of wheelrelated convex polytopes in [24] and characterized these graphs.…”

Section: Introduction and Preliminarymentioning

confidence: 99%

On Resolvability Parameters of Some Wheel-Related Graphs

Yang

Rafiullah

Siddiqui

et al. 2019

Journal of Chemistry

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Let G=V,E be a simple connected graph, w∈V be a vertex, and e=uv∈E be an edge. The distance between the vertex w and edge e is given by de,w=mindw,u,dw,v, A vertex w distinguishes two edges e1, e2∈E if dw,e1≠dw,e2. A set S is said to be resolving if every pair of edges of G is distinguished by some vertices of S. A resolving set with minimum cardinality is the basis for G, and this cardinality is the edge metric dimension of G, denoted by edimG. It has already been proved that the edge metric dimension is an NP-hard problem. The main objective of this article is to study the edge metric dimension of some families of wheel-related graphs and prove that these families have unbounded edge metric dimension. Moreover, the results are compared with the metric dimension of these graphs.

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