Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 11 4 2026 12 01 Breaking Symmetry in Graphs by Resolving Sets 1181 1195 15133 10.22049/cco.2026.30820.2632 EN Nasrin Soltankhah Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran Meysam Korivand Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran Sandi Klavžar Faculty of Mathematics and Physics, University of Ljubljana, Slovenia Journal Article 2025 07 22 Let $dim(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le dim(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars attain the bound, and among connected unicyclic graphs such graphs are $t$-cycles for $t\in \{3,4,5\}$. It is shown that for any $1\leq n< m$, there exists a graph $G$ with $D(G)=n$ and ${\rm dim}(G)=m$. Using the bound $D(G) \le dim(G)+1$, graphs with $D(G) = n(G)-2$ are classified.  https://comb-opt.azaruniv.ac.ir/article_15133_865c85d6dc7a3f8f56b24284c6611b43.pdf