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Gem- And Co-Gem-Free Graphs Have Bounded Clique-Width

The P4 is the induced path of four vertices. The gem consists of a P4 with an additional universal vertex being completely adjacent to the P4, and the co-gem is its complement graph. Gem- and co-gem-free graphs generalize the popular class of cographs (i. e. P4-free graphs). The tree structure and algebraic generation of cographs has been crucial for several concepts of graph decomposition such as modular and homogeneous decomposition. Moreover, it is fundamental for the recently introduced concept of clique-width of graphs which extends the famous concept of treewidth. It is well-known that the cographs are exactly those graphs of clique-width at most 2. In this paper, we show that the clique-width of gem- and co-gem-free graphs is at most 16.