Lettericity of graphs: an FPT algorithm and a bound on the size of obstructions

TL;DR

This paper introduces a fixed-parameter tractable algorithm for recognizing graphs with bounded lettericity and establishes an upper bound on the size of minimal obstructions, advancing understanding of graph structure.

Contribution

The paper provides the first constructive recognition algorithm for graphs of bounded lettericity and bounds the size of minimal obstructions, which were previously unknown.

Findings

Developed an $f(k)n^3$ time recognition algorithm for graphs with lettericity at most $k$.

Proved that minimal graphs with lettericity exceeding $k$ have at most $2^{O(k^2 ext{log}k)}$ vertices.

Established structural bounds on obstructions related to lettericity.

Abstract

Lettericity is a graph parameter responsible for many attractive structural properties. In particular, graphs of bounded lettericity have bounded linear clique-width and they are well-quasi-ordered by induced subgraphs. The latter property implies that any hereditary class of graphs of bounded lettericity can be described by finitely many forbidden induced subgraphs. This, in turn, implies, in a non-constructive way, polynomial-time recognition of such classes. However, no constructive algorithms and no specific bounds on the size of forbidden graphs are available up to date. In the present paper, we develop an algorithm that recognizes $n$-vertex graphs of lettericity at most $k$ in time $f(k)n^3$ and show that any minimal graph of lettericity more than $k$ has at most $2^{O(k^2\log k)}$ vertices.