Some resolving parameters with the minimum size for two specific graphs
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866929439147819008 |
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| author | Zafari, Ali Alikhani, Saeid |
| author_facet | Zafari, Ali Alikhani, Saeid |
| contents | A resolving set for a graph $G$ is a set of vertices $Q = \{q_1, ..., q_k\}$ such that, for all $p\in V(G)$ the $k$-tuple $(d(p, q_1), ..., d(p, q_k ))$ uniquely determines $p$, where $d(p, q_i)$ is considered as the minimum length of a shortest path from $p$ to $q_i$ in graph $G$. In this paper, we consider the computational study of some resolving sets with the minimum size for the $m$-cylinder graph $(C_n\Box P_k)\Box P_m$. The Boolean lattice $BL_n$, $n\geq 1$, is the graph whose vertex set is the set of all subsets of $[n]=\{1,2,...,n\}$, where two subsets $X$ and $Y$ are adjacent if their symmetric difference has precisely one element. In the graph $BL_n$, the layer $L_i$ is the family of $i$-subsets of $[n]$. The subgraph $BL_n(i,i+1)$ is the subgraph of $BL_n$ induced by layers $L_i$ and $L_{i+1}$. Usually the graph $BL_n(1,2)$ is denoted by $H(n)$. We study the minimum size of a resolving set, doubly resolving set and strong resolving set for the graph $L(n)$, which is the line graph of $H(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_15560 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Some resolving parameters with the minimum size for two specific graphs Zafari, Ali Alikhani, Saeid Combinatorics 05C12, 05C76 A resolving set for a graph $G$ is a set of vertices $Q = \{q_1, ..., q_k\}$ such that, for all $p\in V(G)$ the $k$-tuple $(d(p, q_1), ..., d(p, q_k ))$ uniquely determines $p$, where $d(p, q_i)$ is considered as the minimum length of a shortest path from $p$ to $q_i$ in graph $G$. In this paper, we consider the computational study of some resolving sets with the minimum size for the $m$-cylinder graph $(C_n\Box P_k)\Box P_m$. The Boolean lattice $BL_n$, $n\geq 1$, is the graph whose vertex set is the set of all subsets of $[n]=\{1,2,...,n\}$, where two subsets $X$ and $Y$ are adjacent if their symmetric difference has precisely one element. In the graph $BL_n$, the layer $L_i$ is the family of $i$-subsets of $[n]$. The subgraph $BL_n(i,i+1)$ is the subgraph of $BL_n$ induced by layers $L_i$ and $L_{i+1}$. Usually the graph $BL_n(1,2)$ is denoted by $H(n)$. We study the minimum size of a resolving set, doubly resolving set and strong resolving set for the graph $L(n)$, which is the line graph of $H(n)$. |
| title | Some resolving parameters with the minimum size for two specific graphs |
| topic | Combinatorics 05C12, 05C76 |
| url | https://arxiv.org/abs/2103.15560 |