Algorithmic methods of finite discrete structures. Graph clique problem

TL;DR

This paper introduces a novel algorithm for finding maximum cliques in nonseparable graphs using cycle representations and vector analysis, offering a new approach to solving the clique problem.

Contribution

It presents a new cycle-based algorithm for maximum clique detection in nonseparable graphs, with analysis of its computational complexity.

Findings

The algorithm effectively identifies maximum cliques.

Cycle vector analysis aids in clique detection.

Computational complexity is thoroughly analyzed.

Abstract

The monography presents a new algorithm for finding the clique of maximal length in a nonseparable graph. The algorithm is based on the properties of the representation of a clique as a subset of the set of cycles with a length of three, the ring sum of which is an empty set. As a result of selecting the cycles of the length of three, two vectors are formed: the vector of cycles passing through the edges and the vector of cycles passing through the vertices. The numerical values of the components of these vectors determine the weights of the vertices and edges. The iterative process of constructing the set of vectors of cycles passing through the edges allows identifying the main vector of cycles passing through the edges. In turn, the construction of the main vector allows finding the clicks of the graph. The computational complexity of the presented algorithm is analyzed.