%0 Journal Article
%T Unicyclic graphs with non-isolated resolving number $2$
%J Transactions on Combinatorics
%I University of Isfahan
%Z 2251-8657
%A Jannesari, Mohsen
%D 2023
%\ 06/01/2023
%V 12
%N 2
%P 73-78
%! Unicyclic graphs with non-isolated resolving number $2$
%K non-isolated resolving sets
%K Unicyclic graphs
%K Metric dimension
%R 10.22108/toc.2022.129790.1880
%X 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.
%U https://toc.ui.ac.ir/article_26496_8db320090b312f97cc28e3165deb95d9.pdf