On Modal m-Calculus over Finite Graphs with Bounded Strongly Connected Components

For every positive integer k we consider the class SCCk of all finite graphs whose strongly connected components have size at most k. We show that for every k. the Modal ^-Calculus fixpoint hierarchy on SCCk collapscs to the level A2 = IT2 f i £2- but not to Comp(X\. 11] j (compositions of formulas of level Ei and 111). This contrasts with the class of all graphs, where A2 = CompCL\. 11] j.