Succinct Data Structures for Series-Parallel, Block-Cactus and 3-Leaf Power Graphs

TL;DR

This paper introduces space-efficient data structures for specific graph classes that enable fast navigation queries, achieving optimal space and query performance even without known lower bounds.

Contribution

It provides the first succinct data structures with optimal query support for series-parallel, block-cactus, and 3-leaf power graphs, extending previous work to multigraphs.

Findings

Achieves optimal space for the graph classes.

Supports degree, adjacency, and neighborhood queries efficiently.

Extends prior work to series-parallel multigraphs.

Abstract

We design succinct encodings of {\it series-parallel, block-cactus} and {\it 3-leaf power} graphs while supporting the basic navigational queries such as degree, adjacency and neighborhood {\it optimally} in the RAM model with logarithmic word size. One salient feature of our representation is that it can achieve optimal space even though the exact space lower bound for these graph classes is not known. For these graph classes, we provide succinct data structures with optimal query support for the first time in the literature. For series-parallel multigraphs, our work also extends the works of Uno et al. (Disc. Math. Alg. and Appl., 2013) and Blelloch and Farzan (CPM, 2010) to produce optimal bounds.