All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs

TL;DR

This paper presents efficient algorithms for listing all maximal independent sets in sparse graphs and introduces dynamic data structures for maintaining and querying vertex sets, with applications to graph enumeration and domination problems.

Contribution

It introduces polynomial-time algorithms for listing maximal independent sets in sparse graphs and develops new dynamic data structures for vertex set maintenance and domination queries.

Findings

Constant time per set in bounded degree graphs

O(n) time per set in minor-closed graph families

Subquadratic time per set in general sparse graphs

Abstract

We describe algorithms, based on Avis and Fukuda's reverse search paradigm, for listing all maximal independent sets in a sparse graph in polynomial time and delay per output. For bounded degree graphs, our algorithms take constant time per set generated; for minor-closed graph families, the time is O(n) per set, and for more general sparse graph families we achieve subquadratic time per set. We also describe new data structures for maintaining a dynamic vertex set S in a sparse or minor-closed graph family, and querying the number of vertices not dominated by S; for minor-closed graph families the time per update is constant, while it is sublinear for any sparse graph family. We can also maintain a dynamic vertex set in an arbitrary m-edge graph and test the independence of the maintained set in time O(sqrt m) per update. We use the domination data structures as part of our enumeration algorithms.