Detecting 2-joins faster

TL;DR

This paper introduces a significantly faster algorithm for detecting 2-joins in graphs, reducing the complexity from previous methods and enabling quicker algorithms for graph decomposition tasks.

Contribution

The paper presents an $O(n^2m)$-time algorithm for detecting 2-joins and non-path 2-joins, improving upon the previous $O(n^3m)$ and $O(n^4m)$ methods.

Findings

Speeds up 2-join detection algorithms

Reduces complexity from cubic and quartic to quadratic

Enables faster graph decomposition processes

Abstract

2-joins are edge cutsets that naturally appear in the decomposition of several classes of graphs closed under taking induced subgraphs, such as balanced bipartite graphs, even-hole-free graphs, perfect graphs and claw-free graphs. Their detection is needed in several algorithms, and is the slowest step for some of them. The classical method to detect a 2-join takes $O(n^3m)$ time where $n$ is the number of vertices of the input graph and $m$ the number of its edges. To detect \emph{non-path} 2-joins (special kinds of 2-joins that are needed in all of the known algorithms that use 2-joins), the fastest known method takes time $O(n^4m)$. Here, we give an $O(n^2m)$-time algorithm for both of these problems. A consequence is a speed up of several known algorithms.