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The Metric Dimension of Unicyclic Graphs

Christopher Poisson1, Ping Zhang1
1Department of Mathematics and Statistics Western Michigan University Kalamazoo, MI 49008

Abstract

For an ordered set \(W = \{w_1, w_2, \ldots, w_k\}\) of vertices and a vertex \(v\) in a graph \(G\), the representation of \(v\) with respect to \(W$ is the \(k\)-vector \(r(v|W) = (d(v, w_1), d(v, w_2), \ldots, d(v, w_k))\), where \(d(x,y)\) represents the distance between the vertices \(x\) and \(y\). The set \(W\) is a resolving set for \(G\) if distinct vertices of \(G\) have distinct representations. A resolving set containing a minimum number of vertices is called a basis for \(G\) and the number of vertices in a basis is the (metric) dimension \(\dim G\). A connected graph is unicyclic if it contains exactly one cycle. For a unicyclic graph \(G\), tight bounds for \(\dim G\) are derived. It is shown that all numbers between these bounds are attainable as the dimension of some unicyclic graph.