层太阳图和层太阳图的线图的最小双可解集和强度量维数的计算
Computing Minimal Doubly Resolving Sets and the Strong Metric Dimension of the Layer Sun Graph and the Line Graph of Layer Sun Graph
摘要：
设$G$是一个阶至少为2的有限连通图，其顶点集为$V(G)$，边集为$E (G)$ .图$G$的顶点集$S$是$G$的双可解集，如果$G$的每两个不同的顶点都被$S$的某两个顶点双可解.图$G$的最小顶点双可分解集是具有最小基数的双可分解集，用$ψ(G)$表示。本文首先构造了一类阶为$2n+ Σ_{r=1}^{k-2}nm^{r}$的图，记为$LSG(n,m, k)$，并称这类图为带参数$n$，$m$和$k$的层Sun图.此外，我们还计算了层Sun图$LSG(n,m, k)$和层Sun图$LSG(n,m, k)$的线图的最小双可解集和强度量维数.
英文摘要：
Let $G$ be a finite, connected graph of order of at least 2, with vertex set $V(G)$ and edge set $E (G)$. A set $S$ of vertices of the graph $G$ is a doubly resolving set for $G$ if every two distinct vertices of $G$ are doubly resolved by some two vertices of $S$. The minimal doubly resolving set of vertices of graph $G$ is a doubly resolving set with minimum cardinality and is denoted by $ψ(G)$. In this paper, first, we construct a class of graphs of order $2n+ Σ_{r=1}^{k-2}nm^{r}$, denoted by $LSG(n,m, k)$, and call these graphs as the layer Sun graphs with parameters $n$, $m$ and $k$. Moreover, we compute minimal doubly resolving sets and the strong metric dimension of layer Sun graph $LSG(n,m, k)$ and the line graph of the layer Sun graph $LSG(n,m, k)$.
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