树在$L_{\infty}$度量下的SIG维数
On the SIG dimension of trees under $L_{\infty}$ metric
摘要：
我们研究了树在$L_{\infty}$度量下的$SIG$维数，并回答了Michael和Quint（Discrete Applied Mathematics：127，pages 447-460，2003）提出的一个公开问题.假设$T$是一棵至少有两个顶点的树。对于每个$v\in V(T)$，让叶度$(v)$表示$v$的叶子邻居的数量。我们将最大叶度定义为$α(T) = \max_{x \in V(T)}$叶度$(x)$。令$S = \{v\in V(T) |$叶度$(v) = α\}$。如果是$|S| = 1$，则定义$β(T) = α(T) - 1$。否则定义$β(T) = α(T)$。我们表明，对于一棵树$T$，$SIG_\infty(T) = \lceil \log_2(β+ 2)\rceil$其中$β= β(T)$，提供$β$不是$2^k - 1$的形式，对于一些正整数$k \geq 1$。如果是$β= 2^k - 1$，则为$SIG_\infty (T) \in \{k, k+1\}$。我们证明了这两个值都是可能的。
英文摘要：
We study the $SIG$ dimension of trees under $L_{\infty}$ metric and answer an open problem posed by Michael and Quint (Discrete Applied Mathematics: 127, pages 447-460, 2003). Let $T$ be a tree with atleast two vertices. For each $v\in V(T)$, let leaf-degree$(v)$ denote the number of neighbours of $v$ that are leaves. We define the maximum leaf-degree as $α(T) = \max_{x \in V(T)}$ leaf-degree$(x)$. Let $S = \{v\in V(T) |$ leaf-degree$(v) = α\}$. If $|S| = 1$, we define $β(T) = α(T) - 1$. Otherwise define $β(T) = α(T)$. We show that for a tree $T$, $SIG_\infty(T) = \lceil \log_2(β+ 2)\rceil$ where $β= β(T)$, provided $β$ is not of the form $2^k - 1$, for some positive integer $k \geq 1$. If $β= 2^k - 1$, then $SIG_\infty (T) \in \{k, k+1\}$. We show that both values are possible.
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