Geodesic
On unicyclic graphs of metric dimension~$2$
Algebra and discrete mathematics, Tome 23 (2017) no. 2, pp. 216-222

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A metric basis $S$ of a graph $G$ is the subset of vertices of minimum cardinality such that all other vertices are uniquely determined by their distances to the vertices in $S$. The metric dimension of a graph $G$ is the cardinality of the subset $S$. A unicyclic graph is a graph containing exactly one cycle. The construction of a knitting unicyclic graph is introduced. Using this construction all unicyclic graphs with two main vertices and metric dimensions $2$ are characterized.
Keywords: metric dimensions
Mots-clés : distance, unicyclic graph. 
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Marharyta Dudenko; Bogdana Oliynyk. On unicyclic graphs of metric dimension~$2$. Algebra and discrete mathematics, Tome 23 (2017) no. 2, pp. 216-222. http://geodesic.mathdoc.fr/item/ADM_2017_23_2_a3/