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Clique-width is a well-studied graph parameter. For graphs of bounded clique-width, many problems that are NP-hard in general can be polynomial-time solvable. The fact motivates several studies to investigate whether the clique-width of graphs in a certain class is bounded or not. We focus on unigraphs, that is, graphs that are uniquely determined by their degree sequences up to isomorphism. We show that every unigraph has clique-width at most 4. It follows that many problems that are NP-hard in general are polynomial-time solvable for unigraphs.