Chordal co-gem-free and (P5,gem)-free graphs have bounded clique-width

It is well known that the clique-width of chordal gem-free graphs (also known as ptolemaic graphs), as a subclass of distance-hereditary graphs, is at most 3. Hereby, the gem consists of a P4 plus a vertex being completely adjacent to the P4, and the co-gem is its complement graph. On the other hand, unit interval graphs being another important subclass of chordal graphs, have unbounded clique-width. In this note, we show that, based on certain tree structure and module properties, chordal co-gem-free graphs have clique-width at most eight. By a structure result for (P5,gem)-free graphs, this implies bounded clique-width for this class as well. Moreover, known results on unbounded clique-width of certain grids and of split graphs imply that the gem and the co-gem are the only one-vertex P4 extension H such that chordal H-free graphs have bounded clique-width.