We consider the problem of designing succinct navigational oracles, i.e., succinct data structures supporting basic navigational queries such as degree, adjacency, and neighborhood efficiently for intersection graphs on a circle, which include graph classes such as {\it circle graphs}, {\it $k$-polygon-circle graphs}, {\it circle-trapezoid graphs}, {\it trapezoid graphs}. The degree query reports the number of incident edges to a given vertex, the adjacency query asks if there is an edge between two given vertices, and the neighborhood query enumerates all the neighbors of a given vertex. We first prove a general lower bound for these intersection graph classes and then present a uniform approach that lets us obtain matching lower and upper bounds for representing each of these graph classes. More specifically, our lower bound proofs use a unified technique to produce tight bounds for all these classes, and this is followed by our data structures which are also obtained from a unified representation method to achieve succinctness for each class. In addition, we prove a lower bound of space for representing {\it trapezoid} graphs and give a succinct navigational oracle for this class of graphs.

中文翻译：

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圆上相交图族的简洁导航预言

我们考虑了设计简洁导航预言机的问题，即为圆上的交点图有效地支持基本导航查询（如度、邻接和邻域）的简洁数据结构，其中包括诸如{\it circle graphs}、{\它 $k$-polygon-circle graphs}, {\it circle-trapezoid graphs}, {\it trapezoid graphs}。度查询报告给定顶点的事件边数，邻接查询询问两个给定顶点之间是否存在边，邻域查询枚举给定顶点的所有邻居。我们首先证明这些交集图类的一般下界，然后提出一种统一的方法，让我们获得匹配的下界和上限来表示这些图类中的每一个。进一步来说，我们的下界证明使用统一的技术为所有这些类生成紧边界，然后是我们的数据结构，这些数据结构也是从统一表示方法获得的，以实现每个类的简洁性。此外，我们证明了表示 {\it trapezoid} 图的空间下界，并为此类图提供了一个简洁的导航预言。