论文标题
分区维度和链循环的强度度
Partition dimension and strong metric dimension of chain cycle
论文作者
论文摘要
让$ g $为带顶点套装$ v（g）$的连接图，而边缘集$ e（g）$。对于订购的$ k $ - 分区$π= \ {q_1，\ ldots，q_k \} $的$ v（g）$，相对于$π$是$ k $ - $ k $ - vectors $ r（v |π）= $ r（v |π）=（v，q_1），d（v，q_1），dd（v，q_1）， $ d（v，q_i）$是$ v $和$ q_i $之间的距离。如果$ r（u |π）\ neq r（v |π）$，则分区$π$是一个分区分区，对于每对不同的顶点$ u，v \ in v（g）$。 $ v（g）$的最低$ k $是$ g $的分区维度。如果$ u $属于最短的$ v-w $ path或$ v $，则在v（g）$中的顶点$ w \ in v（g）$ u，v \ in v（g）$，属于最短的$ w $ path。订购的集合$ w = \ {w_ {1}，\ ldots，w_ {t} \} \ subseteq v（g）$是$ g $的强大解决方案，如果对于每两个不同的dertices $ u $ u $ u $ u $ u $ u $ u $ u $ u $ u $和$ v $ g $强大的度量基础是$ g $，是一组强大的解决方案。强度强度的基数称为$ g $的强度度量尺寸。在本文中，我们确定了由偶数循环构成的链循环的分区维度和强度的度量尺寸以及由奇数循环构成的链循环。
Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. For an ordered $k$-partition $Π=\{Q_1,\ldots,Q_k\}$ of $V(G)$, the representation of a vertex $v \in V(G)$ with respect to $Π$ is the $k$-vectors $r(v|Π)=(d(v,Q_1),\ldots,d(v,Q_k))$, where $d(v,Q_i)$ is the distance between $v$ and $Q_i$. The partition $Π$ is a resolving partition if $r(u|Π)\neq r(v|Π)$, for each pair of distinct vertices $u,v \in V(G)$. The minimum $k$ for which there is a resolving $k$-partition of $V(G)$ is the partition dimension of $G$. A vertex $w\in V(G)$ strongly resolves two distinct vertices $u,v \in V(G)$ if $u$ belongs to a shortest $v-w$ path or $v$ belongs to a shortest $u-w$ path. An ordered set $W=\{w_{1},\ldots, w_{t}\}\subseteq V(G)$ is a strong resolving set for $G$ if for every two distinct vertices $u$ and $v$ of $G$ there exists a vertex $w\in W$ which strongly resolves $u$ and $v$. A strong metric basis of $G$ is a strong resolving set of minimal cardinality. The cardinality of a strong metric basis is called strong metric dimension of $G$. In this paper, we determine the partition dimension and strong metric dimension of a chain cycle constructed by even cycles and a chain cycle constructed by odd cycles.
