We consider the problem of deciding whether a k-colored graph can be completed to have a given property. We establish that, when k is not fixed, the completion problem for circular-arc graphs, even unit or proper circular-arc graphs, is NP-complete. When k is fixed, in the case of completion to circular-arc graphs and Helly circular-arc graphs, we fully classify the complexities of the problems. We also show that deciding whether a 3-colored graph can be completed to be strongly chordal can be done in O(n~2) time. As a corollary of our results, the sandwich problem for Helly circular-arc graphs is NP-complete.
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