A graph is "claw-free" if no vertex has three pairwise nonadjacent neighbours. Line
graphs are claw-free, and there has been a great deal of work on extending various
theorems about line graphs to claw-free graphs. Claw-free graphs are more general
than line graphs, but how much more general are they?

A graph is "claw-free" if no vertex has three pairwise nonadjacent neighbours. Line
Certainly there are claw-free graphs that are not line graphs; for instance, the
icosahedron is claw-free, and so is the Schlafli graph, and so is every graph with
stability number at most two. Another type of claw-free graph is made as follows:
arrange vertices in a circular order, choose some intervals from this circular order,
and make two vertices adjacent if and only if one of these intervals contains them
both. In general none of these examples are line graphs.

A graph is "claw-free" if no vertex has three pairwise nonadjacent neighbours. Line
In joint work with Maria Chudnovsky, we found an explicit construction for all claw-free
graphs. We showed that there are a few "basic" types, such as those described above, and
every claw-free graph can be built starting from graphs of these types by piecing them
together using simple composition operations. We explain this work.