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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.

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