# Succinct Data Structures for Families of Interval Graphs

## TL;DR

This paper introduces space-efficient data structures for interval graphs that support various queries in optimal time, answering open questions about the minimal space needed for such representations.

## Contribution

It presents the first succinct data structures for interval graphs supporting multiple queries in optimal time, and extends these techniques to related graph classes.

## Key findings

- Achieved optimal query times for degree, adjacency, neighborhood, and shortest path queries.
- Proved a tight lower bound of n log n - 2n log log n bits for unlabeled interval graphs.
- Developed data structures for various interval graph variants with efficient query support.

## Abstract

We consider the problem of designing succinct data structures for interval graphs with $n$ vertices while supporting degree, adjacency, neighborhood and shortest path queries in optimal time in the $\Theta(\log n)$-bit word RAM model. The degree query reports the number of incident edges to a given vertex in constant time, the adjacency query returns true if there is an edge between two vertices in constant time, the neighborhood query reports the set of all adjacent vertices in time proportional to the degree of the queried vertex, and the shortest path query returns a shortest path in time proportional to its length, thus the running times of these queries are optimal. Towards showing succinctness, we first show that at least $n\log{n} - 2n\log\log n - O(n)$ bits are necessary to represent any unlabeled interval graph $G$ with $n$ vertices, answering an open problem of Yang and Pippenger [Proc. Amer. Math. Soc. 2017]. This is augmented by a data structure of size $n\log{n} +O(n)$ bits while supporting not only the aforementioned queries optimally but also capable of executing various combinatorial algorithms (like proper coloring, maximum independent set etc.) on the input interval graph efficiently. Finally, we extend our ideas to other variants of interval graphs, for example, proper/unit interval graphs, k-proper and k-improper interval graphs, and circular-arc graphs, and design succinct/compact data structures for these graph classes as well along with supporting queries on them efficiently.