Breaking symmetry in graphs by resolving sets

Soltankhah, Nasrin; Korivand, Meysam; Klavžar, Sandi

doi:10.22049/cco.2026.30820.2632

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	Breaking symmetry in graphs by resolving sets
Communications in Combinatorics and Optimization
مقاله 11، دوره 11، شماره 4، اسفند 2026، صفحه 1181-1195 اصل مقاله (415.74 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2026.30820.2632
نویسندگان
Nasrin Soltankhah^* ¹؛ Meysam Korivand²؛ Sandi Klavžar³
¹Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran
²Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran
³Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
چکیده
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.
کلیدواژه‌ها
resolving set؛ metric dimension؛ distinguishing number؛ twin graph؛ almost asymmetric graph

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Breaking symmetry in graphs by resolving sets