Suppose I want to define the class of claw and co-claw free graphs? The graphs G such that neither G nor its complement contains an induced claw ($K\_{1,3}$). 

Do these graphs have any kind of interesting structure besides what is already implicit? What would good examples be? So far I can think of cycles, $K\_n$ minus an edge, cocktail party graphs (<http://mathworld.wolfram.com/CocktailPartyGraph.html>). Also the house graph ($C\_5$ plus an edge), and a few other small graphs. (And their complements of course).