Cyclic stability of a digraph order ($\alpha (\mathcal{C})$)

The cyclic stability is the largest size of an independent set appearing consecutively in a representative of a coherent cyclic order of a digraph. It is at most the ordinary independence number. A set with no forward path between its distinct vertices can be moved into a consecutive block by swapping nonadjacent vertices. Choose a representative minimizing the span of the set; a vertex in a gap can be moved left or right past the block, contradicting minimality unless there are no gaps.