Breaking Symmetry in Graphs by Resolving Sets
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866909675306352640 |
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| author | Korivand, Meysam Soltankhah, Nasrin Klavžar, Sandi |
| author_facet | Korivand, Meysam Soltankhah, Nasrin Klavžar, Sandi |
| contents | Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ 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 $D(G) \le {\rm dim}(G)+1$, graphs with $D(G) = n(G)-2$ are classified. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15781 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Breaking Symmetry in Graphs by Resolving Sets Korivand, Meysam Soltankhah, Nasrin Klavžar, Sandi Combinatorics Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ 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 $D(G) \le {\rm dim}(G)+1$, graphs with $D(G) = n(G)-2$ are classified. |
| title | Breaking Symmetry in Graphs by Resolving Sets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.15781 |