Graph-structural dependence of the \(\tilde{\zeta}\) hierarchy

Determine whether the hierarchy \(\tilde{\zeta}^{(r)}(G)\) depends only on the stability number, graph size, and level, or whether it captures additional structural properties of the graph.

Background

The linear hierarchy $\tilde{\zeta}^{(r)}(G)$ strengthens the previously studied $\zeta^{(r)}(G)$ hierarchy. The latter depends only on $\alpha(G)$, the number of vertices, and the level, so it cannot distinguish graphs with the same such parameters. The paper explicitly asks whether the new hierarchy has the same limitation or reflects finer graph structure.