R. Servedio and Y-N. Yeh.

The study of the preimage problem of an endofunction on circular compositions is motivated by the study of coloring circular-arc graphs. In this paper we establish a 1-1 correspondence between preimages of a given circular composition $S$ and proper $S$-sequences, and also provide a necessary and sufficient condition for a sequence of subsets of the natural numbers to be a proper $S$-sequence for some circular composition $S$.

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