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Metric Dimension and Secure Metric Dimension of Some Graphs |
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PP: 1307-1316 |
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doi:10.18576/amis/190606
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Author(s) |
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Ashraf Elrokh,
Eman S. Almotairi,
Nabawia Elramly,
Ebitsam Mostafa,
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Abstract |
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| The metric dimension of a graph is the smallest number of vertices from which the vector of distances to every vertex in the graph is unique. A resolving set S of G with the minimum cardinality is a metric basis of G, and |S| is the metric dimension of G. A resolving set S is secure if for any v ∈ V − S, there exists x ∈ S such that (S−{x})∪{v} is a resolving set. For some graphs, the value of the resolving set of the graph is determined and defined, the value of the secure resolving number is determined and defined. The results show that different graph families have different Secure resolving set , we will demonstrate that the resolving set dimension and the secure resolving set dimension of including the total graph , the pendent edge graph , the tadpole graph , the open diagonal ladder graph and the bridge graph. This paper investigates the metric and secure metric dimensions of several graph families, including the total graph, pendent edge graph, tadpole graph, open diagonal ladder graph, and bridge graph. For each graph type, we determine the minimal resolving sets and analyze their resolving sets and analyze their structural characteristics. The results reveal notable differences in resolving behavior across these graph classes, offering insights relevant to applications in network discovery, combinatorial optimization, and pattern recognition. |
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