Stability of Special Graph Classes

TL;DR

This paper investigates the computational complexity of stability problems in various special graph classes, providing polynomial-time algorithms for cases where the problem is generally hard.

Contribution

It extends prior work by Frei et al. to develop polynomial algorithms for stability problems on specific graph classes.

Findings

Polynomial algorithms for stability in graphs without edges and complete graphs

Efficient solutions for paths, trees, forests, bipartite graphs, and co-graphs

Broader understanding of stability problem complexity in special graph classes

Abstract

Frei et al. [6] showed that the problem to decide whether a graph is stable with respect to some graph parameter under adding or removing either edges or vertices is $\Theta_2^{\text{P}}$-complete. They studied the common graph parameters $\alpha$ (independence number), $\beta$ (vertex cover number), $\omega$ (clique number), and $\chi$ (chromatic number) for certain variants of the stability problem. We follow their approach and provide a large number of polynomial-time algorithms solving these problems for special graph classes, namely for graphs without edges, complete graphs, paths, trees, forests, bipartite graphs, and co-graphs.