Erd\H{o}s and West (Discrete Mathematics'85) considered the class of $n$ vertex intersection graphs which have a {\em $d$-dimensional} {\em $t$-representation}, that is, each vertex of a graph in the class has an associated set consisting of at most $t$ $d$-dimensional axis-parallel boxes. In particular, for a graph $G$ and for each $d \geq 1$, they consider $i_d(G)$ to be the minimum $t$ for which $G$ has such a representation. For fixed $t$ and $d$, they consider the class of $n$ vertex labeled graphs for which $i_d(G) \leq t$, and prove an upper bound of $(2nt+\frac{1}{2})d \log n - (n - \frac{1}{2})d \log(4\pi t)$ on the logarithm of size of the class. In this work, for fixed $t$ and $d$ we consider the class of $n$ vertex unlabeled graphs which have a {\em $d$-dimensional $t$-representation}, denoted by $\mathcal{G}_{t,d}$. We address the problem of designing a succinct data structure for the class $\mathcal{G}_{t,d}$ in an attempt to generalize the relatively recent results on succinct data structures for interval graphs (Algorithmica'21). To this end, for each $n$ such that $td^2$ is in $o(n / \log n)$, we first prove a lower bound of $(2dt-1)n \log n - O(ndt \log \log n)$-bits on the size of any data structure for encoding an arbitrary graph that belongs to $\mathcal{G}_{t,d}$. We then present a $((2dt-1)n \log n + dt\log t + o(ndt \log n))$-bit data structure for $\mathcal{G}_{t,d}$ that supports navigational queries efficiently. Contrasting this data structure with our lower bound argument, we show that for each fixed $t$ and $d$, and for all $n \geq 0$ when $td^2$ is in $o(n/\log n)$ our data structure for $\mathcal{G}_{t,d}$ is succinct. As a byproduct, we also obtain succinct data structures for graphs of bounded boxicity (denoted by $d$ and $t = 1$) and graphs of bounded interval number (denoted by $t$ and $d=1$) when $td^2$ is in $o(n/\log n)$.

翻译：Erd\H{o}s 与 West (Discrete Mathematics'85) 研究了具有 {\em $d$ 维} {\em $t$ 表示} 的 $n$ 顶点交图类，即该类中每个图的顶点关联一个由至多 $t$ 个 $d$ 维轴平行盒子组成的集合。特别地，对于图 $G$ 及每个 $d \geq 1$，他们定义 $i_d(G)$ 为使 $G$ 具有此类表示的最小 $t$。针对固定的 $t$ 和 $d$，他们研究了满足 $i_d(G) \leq t$ 的 $n$ 顶点标号图类，并证明了该类大小对数的上界为 $(2nt+\frac{1}{2})d \log n - (n - \frac{1}{2})d \log(4\pi t)$。本文中，针对固定的 $t$ 和 $d$，我们考虑具有 {\em $d$ 维 $t$ 表示} 的 $n$ 顶点无标号图类，记为 $\mathcal{G}_{t,d}$。我们着手为 $\mathcal{G}_{t,d}$ 类设计简洁数据结构，旨在推广近期关于区间图简洁数据结构的成果 (Algorithmica'21)。为此，对于满足 $td^2$ 属于 $o(n / \log n)$ 的每个 $n$，我们首先证明了编码 $\mathcal{G}_{t,d}$ 中任意图的任何数据结构至少需要 $(2dt-1)n \log n - O(ndt \log \log n)$ 比特的下界。随后，我们为 $\mathcal{G}_{t,d}$ 提出了一种 $((2dt-1)n \log n + dt\log t + o(ndt \log n))$ 比特的数据结构，该结构能高效支持导航查询。通过将该数据结构与下界论证对比，我们证明：对于每个固定的 $t$ 和 $d$，以及所有满足 $td^2$ 属于 $o(n/\log n)$ 的 $n \geq 0$，我们为 $\mathcal{G}_{t,d}$ 设计的数据结构是简洁的。作为副产品，当 $td^2$ 属于 $o(n/\log n)$ 时，我们还获得了有界盒图（记为 $d$ 且 $t = 1$）和有界区间数图（记为 $t$ 且 $d=1$）的简洁数据结构。