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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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