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 (C₄,P₄)-free. Since (C₄,P₄)-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 (C₄,P₄,2K₂)-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 P₄, 2K₂, and co-2P₃.