Optimization problem for star covers of graphs without four cycles

TL;DR

This paper investigates optimizing star covers in graphs without four cycles, focusing on minimizing bipartite components and developing algorithms to determine the SNT-rank for such graphs.

Contribution

It introduces a novel approach to star covers emphasizing bipartite components and provides an algorithm for computing the SNT-rank in graphs without four cycles.

Findings

Developed an algorithm for SNT-rank determination in graphs without four cycles.

Highlighted the complexity of optimizing bipartite components in star covers.

Connected the problem to nonnegative trifactorization and SNT-rank concepts.

Abstract

This work presents a study of star covers on graphs. Unlike traditional formulations that minimize the number of stars, our aim is to optimize the number of bipartite components used in the cover. This problem, motivated by a symmetric nonnegative trifactorization of matrices and the SNT-rank of graphs, is in general hard to solve. We consider a family of graphs that do not contain four cycles, and develop an algorithm to determine the SNT-rank of such graphs.