arXiv 1003.48552010-03math.CO

关于笛卡儿积图的划分维数的一个注记

A note on the partition dimension of Cartesian product graphs

Ismael G. Yero, Juan A. Rodriquez-Velazquez

发表信息：: Applied Mathematics and Computation 217 (7) (2010) 3571-3574
DOI：: 10.1016/j.amc.2010.08.038

摘要：

设$G=(V,E)$是一个连通图。两个顶点之间的距离$u,v\in V$，由$d(u, v)$表示，是$G$中最短$u-v$路径的长度。顶点$v\in V$与子集$P\subset V$之间的距离定义为$min\{d(v, x): x \in P\}$，并用$d(v, P)$表示。一个图$G$的顶点的有序划分$\{P_1,P_2, ...,P_t\}$，是$G$的\emph{resolving partition}，如果所有的距离向量$(d(v,P_1),d(v,P_2),...,d(v,P_t))$是不同的。$G$的\emph{partition dimension}，用$pd(G)$表示，是$G$的任何分解分区中的最小集合数。本文研究了笛卡儿积图的划分维数。更精确地说，我们证明了对于所有的连通图对$G, H$，$pd(G\times H)\le pd(G)+pd(H)$和$pd(G\times H)\le pd(G)+dim(H).$，因此，我们证明了$pd(G\times H)\le dim(G)+dim(H)+1.$

英文摘要：

Let $G=(V,E)$ be a connected graph. The distance between two vertices $u,v\in V$, denoted by $d(u, v)$, is the length of a shortest $u-v$ path in $G$. The distance between a vertex $v\in V$ and a subset $P\subset V$ is defined as $min\{d(v, x): x \in P\}$, and it is denoted by $d(v, P)$. An ordered partition $\{P_1,P_2, ...,P_t\}$ of vertices of a graph $G$, is a \emph{resolving partition}of $G$, if all the distance vectors $(d(v,P_1),d(v,P_2),...,d(v,P_t))$ are different. The \emph{partition dimension} of $G$, denoted by $pd(G)$, is the minimum number of sets in any resolving partition of $G$. In this article we study the partition dimension of Cartesian product graphs. More precisely, we show that for all pairs of connected graphs $G, H$, $pd(G\times H)\le pd(G)+pd(H)$ and $pd(G\times H)\le pd(G)+dim(H).$ Consequently, we show that $pd(G\times H)\le dim(G)+dim(H)+1.$

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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