Vol.:(0123456789) Algorithmica https://doi.org/10.1007/s00453-020-00710-w 1 3 Succinct Encodings for Families of Interval Graphs Hüseyin Acan 1  · Sankardeep Chakraborty 2  · Seungbum Jo 3  · Srinivasa Rao Satti 4 Received: 19 June 2019 / Accepted: 2 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020 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. Towards showing succinctness, we frst show that at least n log 2 n - 2n log 2 log 2 n - O(n) bits are necessary to represent any unlabeled inter- val graph G with n vertices, answering an open problem of Yang and Pippenger (Proc Am Math Soc Ser B 4(1):1–3, 2017). This is augmented by a data structure of size n log 2 n + O(n) bits while supporting not only the above queries optimally but also capable of executing various combinatorial algorithms (like proper color- ing, maximum independent set etc.) on interval graphs efciently. Finally, we extend our ideas to other variants of interval graphs, for example, proper/unit interval graphs, k-improper interval graphs, and circular-arc graphs, and design succinct data structures for these graph classes as well along with supporting queries on them efciently. Keywords Space efcient data structures · Succinct encoding · Interval graphs · Proper interval graphs · Unit interval graphs · Unit interval graphs * Seungbum Jo sbjo@chungbuk.ac.kr Hüseyin Acan huseyin.acan@drexel.edu Sankardeep Chakraborty sankardeep.chakraborty@gmail.com Srinivasa Rao Satti ssrao@cse.snu.ac.kr 1 Drexel University, Philadelphia, USA 2 National Institute of Informatics, Tokyo, Japan 3 Chungbuk National University, Cheongju, South Korea 4 Seoul National University, Seoul, South Korea