arXiv 1107.41402011-07-20math.CO

关于线图的度量维数

On the metric dimension of line graphs

Min Feng, Min Xu, Kaishun Wang

备注：: 7 pages

摘要：

设$G$是一个（di）图. $G$中顶点的集合$W$是$G$的一个\emph{resolving set}，如果$G$的每个顶点$u$由它到$W$中所有顶点的距离向量唯一确定。$G$的\emph{metric dimension} $μ(G)$是$G$的所有分解集的最小基数。Cáceres等人的\cite{Ca2}计算了完全二部图的线图的度量维数。最近，Bailey和Cameron \cite{Ba}计算了完全图的线图的度量维数。本文研究$G$的线图$L(G)$的度量维数。特别地，我们证明了除有向圈外的强连通有向图$G$的$μ(L(G))=|E(G)|-|V(G)|$，其中$V(G)$是$G$的顶点集，$E(G)$是$G$的边集.作为推论，给出了de Brujin有向图和Kautz有向图的度量维数.此外，我们证明了$\lceil\log_2Δ(G)\rceil\leqμ(L(G))\leq |V(G)|-2$对于至少有五个顶点的简单连通图$G$，其中$Δ(G)$是$G$的最大度。最后，我们得到了树的线图的度量维数。

英文摘要：

Let $G$ be a (di)graph. A set $W$ of vertices in $G$ is a \emph{resolving set} of $G$ if every vertex $u$ of $G$ is uniquely determined by its vector of distances to all the vertices in $W$. The \emph{metric dimension} $μ(G)$ of $G$ is the minimum cardinality of all the resolving sets of $G$. Cáceres et al. \cite{Ca2} computed the metric dimension of the line graphs of complete bipartite graphs. Recently, Bailey and Cameron \cite{Ba} computed the metric dimension of the line graphs of complete graphs. In this paper we study the metric dimension of the line graph $L(G)$ of $G$. In particular, we show that $μ(L(G))=|E(G)|-|V(G)|$ for a strongly connected digraph $G$ except for directed cycles, where $V(G)$ is the vertex set and $E(G)$ is the edge set of $G$. As a corollary, the metric dimension of de Brujin digraphs and Kautz digraphs is given. Moreover, we prove that $\lceil\log_2Δ(G)\rceil\leqμ(L(G))\leq |V(G)|-2$ for a simple connected graph $G$ with at least five vertices, where $Δ(G)$ is the maximum degree of $G$. Finally, we obtain the metric dimension of the line graph of a tree in terms of its parameters.

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。

摘要：

英文摘要：

相关论文