# Constant Amortized Time Enumeration of Independent Sets for Graphs with Bounded Clique Number

## TL;DR

This paper introduces an efficient algorithm for enumerating independent sets in graphs with bounded clique number, achieving constant amortized time and linear space, applicable to various graph classes.

## Contribution

The paper presents $ exttt{EIS}$, a novel algorithm for non-maximal independent set enumeration with optimal amortized time for graphs with bounded clique number.

## Key findings

- Runs in $O(q)$ amortized time where $q$ is the clique number
- Works correctly even without knowing the exact clique number
- Applicable to various graph classes like triangle-free, planar, and $F$-free graphs

## Abstract

In this study, we address the independent set enumeration problem. Although several efficient enumeration algorithms and careful analyses have been proposed for maximal independent sets, no fine-grained analysis has been given for the non-maximal variant. From the main result, we propose an algorithm $\texttt{EIS}$ for the non-maximal variant that runs in $O(q)$ amortized time and linear space, where $q$ is the clique number, i.e., the maximum size of a clique in an input graph. Note that $\texttt{EIS}$ works correctly even if the exact value of $q$ is unknown. Despite its simplicity, $\texttt{EIS}$ is optimal for graphs with a bounded clique number, such as, triangle-free graphs, planar graphs, bounded degenerate graphs, locally bounded expansion graphs, and $F$-free graphs for any fixed graph $F$, where a $F$-free graph is a graph that has no copy of $F$ as a subgraph.