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The metric dimension of amalgamation of cycles. (English) Zbl 1194.05033
Summary: For an ordered set \(W = \{w_1, w_2, \dots, w_k\}\) of vertices and a vertex \(v\) in a connected graph \(G\), the representation of \(v\) with respect to \(W\) is the ordered \(k\)-tuple \(r(v|W) = (d(v,w_1), d(v,w_2), \dots, d(v,w_k))\) where \(d(x,y)\) represents the distance between the vertices \(x\) and \(y\). The set \(W\) is called a resolving set for \(G\) if every vertex of \(G\) has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for \(G\). The dimension of \(G\), denoted by dim\((G)\) is the number of vertices in a basis of \(G\). Let \(\{G_i\}\) be a finite collection of graphs and each \(G_i\) has a fixed vertex \(v_{oi}\) called a terminal. The amalgamation Amal \(\{G_i, v_{oi}\}\) is formed by taking all of the \(G_i\) ’s and identifying their terminals. In this paper, we determine the metric dimension of amalgamation of cycles.

MSC:
05C12 Distance in graphs
05C38 Paths and cycles
05C76 Graph operations (line graphs, products, etc.)
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