We consider the problem of designing a succinct data structure for {\it path graphs} (which are a proper subclass of chordal graphs and a proper superclass of interval graphs) on $n$ vertices while supporting degree, adjacency, and neighborhood queries efficiently. We provide the following two solutions for this problem: - an $n \log n+o(n \log n)$-bit succinct data structure that supports adjacency query in $O(\log n)$ time, neighborhood query in $O(d \log n)$ time and finally, degree query in $\min\{O(\log^2 n), O(d \log n)\}$ where $d$ is the degree of the queried vertex. - an $O(n \log^2 n)$-bit space-efficient data structure that supports adjacency and degree queries in $O(1)$ time, and the neighborhood query in $O(d)$ time where $d$ is the degree of the queried vertex. Central to our data structures is the usage of the classical heavy path decomposition by Sleator and Tarjan~\cite{ST}, followed by a careful bookkeeping using an orthogonal range search data structure using wavelet trees~\cite{Makinen2007} among others, which maybe of independent interest for designing succinct data structures for other graph classes.

翻译：我们考虑在 $n$ 个顶点上设计一种针对{\it 路径图}（它是弦图的真子类且是区间图的真超类）的简洁数据结构，同时高效支持度查询、邻接查询和邻域查询。针对该问题，我们提供以下两种解决方案：- 一种 $n \log n+o(n \log n)$ 比特的简洁数据结构，支持 $O(\log n)$ 时间的邻接查询、$O(d \log n)$ 时间的邻域查询，以及 $\min\{O(\log^2 n), O(d \log n)\}$ 时间的度查询，其中 $d$ 为查询顶点的度。- 一种 $O(n \log^2 n)$ 比特的空间高效数据结构，支持 $O(1)$ 时间的邻接查询和度查询，以及 $O(d)$ 时间的邻域查询，其中 $d$ 为查询顶点的度。我们数据结构的核心在于利用 Sleator 和 Tarjan~\cite{ST} 的经典重路径分解，随后结合小波树~\cite{Makinen2007} 等技术的正交范围搜索数据结构进行精细的记账管理，这一方法可能为其他图类的简洁数据结构设计提供独立的研究价值。