Succinct Data Structures for Path Graphs and Chordal Graphs Revisited

We enhance space efficient representations of two types of intersection graphs. We refine the data structure for path graphs of Balakrishnan et al. to give a succinct data structure of n log n + o(n log n) bits that supports adjacency test, degree and neighbourhood queries in $O\left( {\frac{{\log n}}{{\log \log n}}} \right)$ time (for neighbourhood queries, this is the amount of time for each neighbour reported). To achieve O(1) query times, we give a data structure using (3 + ε)n log n + o(n log n) bits for any constant ε > 0. Furthermore, we are able to support both the distance and shortest path queries on unweighted path graphs using (2 + ε)n log n+ o(n log n) bits in O(log n/ log log n) time (shortest path uses an additional O(1) time per vertex on the path). This is the first compact distance oracles for path graphs. Turning to chordal graphs, we enhance the succinct data structure of Munro and Wu to reduce all query times including performing adjacency test in O(1) time.