Multiset Metric Dimension of Binomial Random Graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909690373341184 |
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| author | Eide, Austin Pralat, Pawel |
| author_facet | Eide, Austin Pralat, Pawel |
| contents | For a graph $G = (V,E)$ and a subset $R \subseteq V$, we say that $R$ is \textit{multiset resolving} for $G$ if for every pair of vertices $v,w$, the \textit{multisets} $\{d(v,r): r \in R\}$ and $\{d(w,r):r \in R\}$ are distinct, where $d(x,y)$ is the graph distance between vertices $x$ and $y$. The \textit{multiset metric dimension} of $G$ is the size of a smallest set $R \subseteq V$ that is multiset resolving (or $\infty$ if no such set exists). This graph parameter was introduced by Simanjuntak, Siagian, and Vitrík in 2017~\cite{simanjuntak2017multiset}, and has since been studied for a variety of graph families. We prove bounds which hold with high probability for the multiset metric dimension of the binomial random graph $G(n,p)$ in the regime $d = (n-1)p = Θ(n^{x})$ for fixed $x \in (0,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11686 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiset Metric Dimension of Binomial Random Graphs Eide, Austin Pralat, Pawel Combinatorics Discrete Mathematics For a graph $G = (V,E)$ and a subset $R \subseteq V$, we say that $R$ is \textit{multiset resolving} for $G$ if for every pair of vertices $v,w$, the \textit{multisets} $\{d(v,r): r \in R\}$ and $\{d(w,r):r \in R\}$ are distinct, where $d(x,y)$ is the graph distance between vertices $x$ and $y$. The \textit{multiset metric dimension} of $G$ is the size of a smallest set $R \subseteq V$ that is multiset resolving (or $\infty$ if no such set exists). This graph parameter was introduced by Simanjuntak, Siagian, and Vitrík in 2017~\cite{simanjuntak2017multiset}, and has since been studied for a variety of graph families. We prove bounds which hold with high probability for the multiset metric dimension of the binomial random graph $G(n,p)$ in the regime $d = (n-1)p = Θ(n^{x})$ for fixed $x \in (0,1)$. |
| title | Multiset Metric Dimension of Binomial Random Graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2507.11686 |