arXiv 2005.073112020-05-15math.COcs.DM

图的广播维数

Broadcast Dimension of Graphs

Jesse Geneson, Eunjeong Yi

摘要：

在本文中，我们开始研究广播维数，度量维数的一个变种。设$G$是一个顶点集为$V(G)$的图，$d(u,w)$表示$G$中$u-w$测地线的长度。对于$k \ge 1$，设为$d_k(x,y)=\min \{d(x,y), k+1\}$。函数$f: V(G) \rightarrow \mathbb{Z}^+ \cup \{0\}$被称为$G$的解析广播，如果对于任何不同的$x,y \in V(G)$，存在顶点$z \in V(G)$使得$f(z)=i>0$和$d_{i}(x,z) \neq d_{i}(y,z)$。$G$的广播维度$bdim(G)$是$G$的所有解析广播上的$c_f(G)=\sum_{v \in V(G)} f(v)$的最小值，其中$c_f(G)$可以被视为用于解析由图$G$描述的整个网络的发射机（各种强度）的总成本。请注意，如果解析广播的共域被限制为$\{0,1\}$，则$bdim(G)$简化为$adim(G)$（$G$的邻接维度，由Jannesari和Omoomi在2012年引入）。我们确定它的值的循环，路径，和其他家庭的图。我们证明了对于所有阶为$n$的图$G$，都有$bdim(G) = Ω(\log{n})$，并且这个结果是尖锐的，直到一个常数因子。我们证明了$\frac{adim(G)}{bdim(G)}$和$\frac{bdim(G)}{dim(G)}$都可以是任意大的，其中$dim(G)$表示$G$的度量维数。我们还研究了顶点删除对图的邻接维数和广播维数的影响。

英文摘要：

In this paper we initiate the study of broadcast dimension, a variant of metric dimension. Let $G$ be a graph with vertex set $V(G)$, and let $d(u,w)$ denote the length of a $u-w$ geodesic in $G$. For $k \ge 1$, let $d_k(x,y)=\min \{d(x,y), k+1\}$. A function $f: V(G) \rightarrow \mathbb{Z}^+ \cup \{0\}$ is called a resolving broadcast of $G$ if, for any distinct $x,y \in V(G)$, there exists a vertex $z \in V(G)$ such that $f(z)=i>0$ and $d_{i}(x,z) \neq d_{i}(y,z)$. The broadcast dimension, $bdim(G)$, of $G$ is the minimum of $c_f(G)=\sum_{v \in V(G)} f(v)$ over all resolving broadcasts of $G$, where $c_f(G)$ can be viewed as the total cost of the transmitters (of various strength) used in resolving the entire network described by the graph $G$. Note that $bdim(G)$ reduces to $adim(G)$ (the adjacency dimension of $G$, introduced by Jannesari and Omoomi in 2012) if the codomain of resolving broadcasts is restricted to $\{0,1\}$. We determine its value for cycles, paths, and other families of graphs. We prove that $bdim(G) = Ω(\log{n})$ for all graphs $G$ of order $n$, and that the result is sharp up to a constant factor. We show that $\frac{adim(G)}{bdim(G)}$ and $\frac{bdim(G)}{dim(G)}$ can both be arbitrarily large, where $dim(G)$ denotes the metric dimension of $G$. We also examine the effect of vertex deletion on the adjacency dimension and the broadcast dimension of graphs.

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