# Parameterized Aspects of Triangle Enumeration

## TL;DR

This paper conducts a comprehensive parameterized complexity analysis of triangle enumeration in graphs, introducing new algorithms and kernelizations, while also establishing limitations for certain parameters.

## Contribution

It presents novel parameterized algorithms and kernelizations for triangle enumeration, and explores the boundaries of efficiency improvements based on graph parameters.

## Key findings

- New subcubic parameterized algorithms for triangle enumeration
- Kernelization techniques for specific graph parameters
- Negative results indicating limits of parameterized improvements

## Abstract

The task of listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to both theoretical aspects (e.g., lower and upper bounds on running time, adaption to new computational models) as well as practical aspects (e.g. algorithms tuned for large graphs). Motivated by the fact that the worst-case running time is cubic, we perform a systematic parameterized complexity study of triangle enumeration. We provide both positive results (new enumerative kernelizations, "subcubic" parameterized solving algorithms) as well as negative results (presumable uselessness in terms of "faster" parameterized algorithms of certain parameters such as graph diameter). To this end, we introduce new and extend previous concepts.