Letter graphs and modular decomposition

TL;DR

This paper establishes conditions under which classes of graphs with prime graphs of bounded lettericity also have bounded lettericity, linking this property to the absence of large matchings, co-matchings, or stacked paths.

Contribution

It introduces the concept of lettericity in graph classes and characterizes when bounded lettericity extends from prime graphs to entire classes based on forbidden structures.

Findings

Bounded lettericity in prime graphs implies bounded lettericity in the class under certain conditions.

Presence of large matchings, co-matchings, or stacked paths prevents bounded lettericity in the class.

Provides a characterization of graph classes with bounded lettericity based on structural obstructions.

Abstract

We prove that if the prime graphs in a graph class have bounded lettericity, then the entire class has bounded lettericity if and only if it does not contain arbitrary large matchings, co-matchings, or a family of graphs that we call stacked paths.