Unicyclic graphs with non-isolated resolving number $2$

Document Type : Research Paper


Department of Science, Shahreza Campus, University of Isfahan, 86149-56841, Shahreza, Iran



Let $G$ be a connected graph and $W=\{w_1, w_2,\ldots,w_k\}$ be an ordered subset of vertices of $G$. For any vertex $v$ of $G$, the ordered $k$-vector $$r(v|W)=(d(v,w_1), d(v,w_2),\ldots,d(v,w_k))$$ is called the metric representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. A set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct metric representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension denoted by $\dim(G)$. A resolving set $W$ is called a non-isolated resolving set for $G$ if the induced subgraph $\langle W\rangle$ of $G$ has no isolated vertices. The minimum cardinality of a non-isolated resolving set for $G$ is called the non-isolated resolving number of $G$ and denoted by $nr(G)$. The aim of this paper is to find properties of unicyclic graphs that have non-isolated resolving number $2$ and then to characterize all these graphs.


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