Succinct Navigational Oracles for Families of Intersection Graphs on a Circle

TL;DR

This paper develops succinct data structures for efficiently supporting navigational queries on various intersection graph classes on a circle, providing tight bounds and a unified approach for multiple graph types.

Contribution

It introduces a unified method to establish matching lower and upper bounds for succinct representations of intersection graphs on a circle, including trapezoid graphs.

Findings

Established tight bounds for graph representations.

Developed a uniform approach for multiple graph classes.

Provided a succinct oracle for trapezoid graphs.

Abstract

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.