Letter graphs and geometric grid classes of permutations: characterization and recognition

TL;DR

This paper establishes a connection between letter graphs and geometric grid classes of permutations, providing a polynomial-time recognition algorithm for 3-letter graphs based on structural characterization.

Contribution

It introduces the first constructive polynomial-time recognition algorithm for 3-letter graphs, expanding understanding of these graph classes.

Findings

Polynomial-time recognition for 3-letter graphs achieved

Structural characterization of 3-letter graphs developed

Connection between letter graphs and geometric grid classes clarified

Abstract

In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a non-constructive way, a polynomial-time recognition of geometric grid classes of permutations and $k$-letter graphs for a fixed $k$. However, constructive algorithms are available only for $k=2$. In this paper, we present the first constructive polynomial-time algorithm for the recognition of $3$-letter graphs. It is based on a structural characterization of graphs in this class.