Abstract:

A graph is said to be clawfree if it has no induced subgraph 
isomorphic to $K_{1,3}$. Line graphs are one well-known class of 
clawfree graphs, but there others, such as circular arc graphs and 
subgraphs of the Schl\"{a}fli graph.  It has been an open question to 
describe the structure of all clawfree graphs. Recently, in joint work 
with Paul Seymour, we were able to prove that all clawfree graphs can be 
constructed from basic pieces (which include the graphs mentioned above, 
as well as a few other ones) by gluing them together in prescribed ways.  
In this talk we will explain the theorem and  present some examples of 
clawfree graphs that are the "basic pieces" for the general structure.  
We will also describe some new  properties of clawfree graphs, that 
we learned while working on the subject.