Letter graphs and well-quasi-order by induced subgraphs

Given a word w over a nite alphabet and a set of ordered pairs of letters which de ne adjacencies we construct a graph which we call the letter graph of w The lettericity of a graph G is the least size of alphabet permitting to obtain G as a letter graph The set of letter graphs consists of threshold graphs unbounded interval graphs and their complements We determine the lettericity of cycles and bound the lettericity of paths to an interval of length one We show that the class of k letter graphs is well quasi ordered by the induced subgraph relation and that it has a nite set of minimal forbidden induced subgraphs As a consequence k letter graphs can be recognized in polynomial time for any xed k