Letter graphs and well-quasi-order by induced subgraphs

Authors:
Marko Petkovsek
Affiliations:
Faculty of Mathematics and Physics, Department of Mathematics and Mechanics, University of Ljubljana, p.p. 14, Jadranska 19, SI-1000 Ljubljana, Slovenia
Venue:
Discrete Mathematics - Algebraic and topological methods in graph theory
Year:
2002

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Abstract

Given a word w over a finite alphabet and a set of ordered pairs of letters which define adjacencies, we construct a graph which we call the letter graph of w. The lettericity of a graph G is the least size of the alphabet permitting to obtain G as a letter graph. The set of 2-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 finite set of minimal forbidden induced subgraphs. As a consequence, k-letter graphs can be recognized in polynomial time for any fixed k.