Erdős and West (Discrete Mathematics’85) considered the class of n vertex intersection graphs which have a d-dimensional t-representation (also called a t,d −intersection 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 ≥ 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 ) ≤ t , and prove an upper bound of $\left( {2nt + \frac{1}{2}} \right)d\log n - \left( {n - \frac{1}{2}} \right)d\log (4\pi t)$ 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 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). Let ${\mathcal{G}_t}{\text{and}}{\mathcal{G}_d}$ be the class of graphs with bounded interval number and bounded boxicity obtained by setting d = 1 and t = 1 in ${\mathcal{G}_{t,d}}$ , respectively. We have the following results:

We consider the problem of designing space-efficient data structures for unlabelled path graphs with$n$vertices while supporting basic navigational queries such as degree, adjacency, and neighborhood queries efficiently. We provide two solutions for this problem. Our first data structure is succinct and occupies$n\log n+o(n\log n)$bits while answering adjacency query in$O(\log n)$time, and neighborhood and degree queries in$O(d\log^{2}n)$time where$d$is the degree of the queried vertex. Our second data structure answers all these queries faster at the expense of slightly more space. More specifically, it consumes$O(n\log^{2}n)$bits while answering adjacency and degree queries in constant time and neighborhood query in$O(d\log n)$time. Central to our data structures is the usage of the classical heavy path decomposition, followed by a careful bookkeeping using an orthogonal range search data structure among others, which may be of independent interest for designing succinct data structures for other graphs.