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Edge metric dimension of \(k\) multiwheel graph. (English) Zbl 1448.05060

Summary: If we consider \(G=(V,E)\) to be a connected graph, with \(v\in V\) and \(e = u w\in E\), then \(d_G (e,v) = \operatorname{min}\{d_G (u,v), d_G (w,v)\}\) has been defined as the distance between a vertex \(v\) and an edge \(e\). The cardinality of the smallest subset \(S \subseteq V\) which can assign a unique distance vector to every edge of \(G\) is referred to as edge metric dimension (EMD) given by \(\operatorname{edim}\,(G)\). A \(k\) multiwheel graph \(W_{1,n,k}\) is composed of \(k\) cycles \(C_n\) along with a central vertex \(x\) such that \(x\) is adjacent to each of the vertices of \(C_1\) and the corresponding vertices of the two consecutive cycles \(C_i\) and \(C_{i+1}\) are also adjacent for all \(1 \leq i \leq (k-1)\). This article discusses the EMD of the double wheel graph and extends the results to the general \(k\) multiwheel graph.

MSC:

05C12 Distance in graphs
05C76 Graph operations (line graphs, products, etc.)
05C90 Applications of graph theory
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References:

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