Bounded Clique-Width of ($S_{1,2,2}$,Triangle)-Free Graphs

If a graph has no induced subgraph isomorphic to $H1$ or $H2$ then it is said to be ($H1,H2$)-free. Dabrowski and Paulusma found 13 open cases for the question whether the clique-width of ($H1,H2$)-free graphs is bounded. One of them is the class of ($S{1,2,2}$,triangle)-free graphs. In this paper we show that these graphs have bounded clique-width. Thus, also ($P1+2P2$,triangle)-free graphs have bounded clique-width which solves another open problem of Dabrowski and Paulusma. Meanwhile we were informed by Paulusma that in December 2015, Dabrowski, Dross and Paulusma showed that ($S{1,2,2}$,triangle)-free graphs (and some other graph classes) have bounded clique-width.