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：Erdős和West (Discrete Mathematics'85) 研究了具有$d$维$t$表示的$n$个顶点的交图类，其中该图类中每个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$G$和任意$d \geq 1$，他们定义$i_d(G)$为$G$具有此类表示所需的最小$t$值。针对固定的$t$和$d$，他们考虑满足$i_d(G) \leq t$的$n$个顶点标记图类，并证明了该类大小的对数上界为$(2nt+\frac{1}{2})d \log n - (n - \frac{1}{2})d \log(4\pi t)$。本文中，对于固定的$t$和$d$，我们考虑具有$d$维$t$表示的$n$个顶点未标记图类，记为$\mathcal{G}_{t,d}$。为推广区间图简洁数据结构的近期成果(Algorithmica'21)，我们着手设计$\mathcal{G}_{t,d}$类的简洁数据结构。为此，对于满足$td^2$属于$o(n / \log n)$的每个$n$，我们首先证明编码任意属于$\mathcal{G}_{t,d}$的图所需任何数据结构的规模下界为$(2dt-1)n \log n - O(ndt \log \log n)$比特。随后我们提出一个支持高效导航查询的$\mathcal{G}_{t,d}$类数据结构，其规模为$((2dt-1)n \log n + dt\log t + o(ndt \log n))$比特。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性图（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。