Characterizations for co-graphs defined by restricted NLC-width or clique-width operations

In this paper we characterize subclasses of co-graphs defined by restricted NLC-width operations and subclasses of co-graphs defined by restricted clique-width operations. We show that a graph has NLCT-width 1 if and only if it is image-free. Since image-free graphs are exactly trivially perfect graphs, the set of graphs of NLCT-width 1 is equal to the set of trivially perfect graphs, and a recursive definition for trivially perfect graphs follows. Further we show that a graph has linear NLC-width 1 if and only if is image-free. This implies that the set of graphs of linear NLC-width 1 is equal to the set of threshold graphs. We also give forbidden induced subgraph characterizations for co-graphs defined by restricted clique-width operations using image, image, and co-image.