RESEARCH ARTICLE

On unicyclic graphs of metric dimension 2 with vertices of degree 4

M. Dudenko, B. Oliynyk

Department of Mathematics, National University of Kyiv-Mohyla Academy, Skovorody St. 2, Kyiv, 04070, Ukraine

Abstract: We show that if $G$ is a unicyclic graph with metric dimension $2$ and $\{a,b\}$ is a metric basis of $G$ then the degree of any vertex $v$ of $G$ is at most $4$ and degrees of both $a$ and $b$ are at most $2$. The constructions of unispider and semiunispider graphs and their knittings are introduced. Using these constructions all unicyclic graphs of metric dimension $2$ with vertices of degree $4$ are characterized.

Keywords: graph, distance, metric dimension, unicyclic graph.

MSC: 05C12

Received: 14.10.2018
Revised: 18.12.2018

Language: English