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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 Θ(log n)bit 1 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.

Research on space efficient graph algorithms, particularly for st-connectivity, has a long history including the celebrated polynomial time, O(lg n) bits 3 algorithm in undirected graphs by Reingold (JACM 2008), and polynomial time, n/2 Θ( Some of these results were announced in preliminary form in the proceedings of 22nd International Computing and Combinatorics Conference (COCOON 2016), Springer LNCS volume 9797, pages 119-130 [7], and 27th International Symposium on Algorithms and Computation (ISAAC 2016), LIPIcs, volume 64, pages 22:1-22:13 [17]. More specifically, this paper contains the linear time algorithms for DFS and its applications, announced in [7] and [17], along with the linear time algorithms for BFS and a few other graph problems, announced in [7]. 3 We use lg to denote logarithm to the base 2.

We consider space efficient implementations of some classical applications of DFS including the problem of testing biconnectivity and 2-edge connectivity, finding cut vertices and cut edges, computing chain decomposition and st-numbering of a given undirected graph G on n vertices and m edges. Classical algorithms for them typically use DFS and some Ω(lg n) bits 1 of information at each vertex. Building on a recent O(n)-bits implementation of DFS due to Elmasry et al. (STACS 2015) we provide O(n)-bit implementations for all these applications of DFS. Our algorithms take O(m lg c n lg lg n) time for some small constant c (where c ≤ 2). Central to our implementation is a succinct representation of the DFS tree and a space efficient partitioning of the DFS tree into connected subtrees, which maybe of independent interest for designing other space efficient graph algorithms.

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