Finding the connected components of the graph using perturbations of the adjacency matrix

TL;DR

This paper introduces a novel method for finding connected components in graphs by using perturbations of the adjacency matrix and solving related linear algebraic systems, offering a potentially more efficient alternative to traditional graph traversal algorithms.

Contribution

The paper proposes a new approach that leverages perturbations of the adjacency matrix and numerical SLAE solutions to determine graph connectivity, differing from standard BFS methods.

Findings

The proposed method can determine connectivity with no greater complexity than BFS.

It often requires less computational effort than traditional algorithms.

The approach enables using efficient numerical linear algebra techniques for graph analysis.

Abstract

The problem of finding the connected components of a graph is considered. The algorithms addressed to solve the problem are used to solve such problems on graphs as problems of finding points of articulation, bridges, maximin bridge, etc. A natural approach to solving this problem is a breadth-first search, the implementations of which are presented in software libraries designed to maximize the use of the capabi\-lities of modern computer architectures. We present an approach using perturbations of adjacency matrix of a graph. We check wether the graph is connected or not by comparing the solutions of the two systems of linear algebraic equations (SLAE): the first SLAE with a perturbed adjacency matrix of the graph and the second SLAE with~unperturbed matrix. This approach makes it possible to use effective numerical implementations of SLAE solution methods to solve connectivity problems on graphs. Iterations of iterative numerical methods for solving such SLAE can be considered as carrying out a graph traversal. Generally speaking, the traversal is not equivalent to the traversal that is carried out with breadth-first search. An algorithm for finding the connected components of a graph using such a traversal is presented. For any instance of the problem, this algorithm has no greater computational complexity than breadth-first search, and for~most individual problems it has less complexity.