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

TL;DR

This work studies lettericity, the minimum alphabet size needed to realise a graph as a letter graph, and shows constructive, fixed-parameter tractable recognition for fixed $k$ via MSO-definability and Courcelle's framework, with a running time of $f(k)n^3$. It introduces a combinatorial MSO encoding that certifies $k$-letter graphs and proves an explicit single-exponential bound on obstructions, showing any obstruction has at most $2^{O(k^2\log k)}$ vertices. The paper also outlines a dynamic-programming alternative for the recognition problem and discusses implications for well-quasi-ordering and connections to permutation-geometry notions, while highlighting open questions about complexity and tighter obstruction bounds.

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.

Theorems & Definitions (47)

• Lemma 2: Petkovsek2002
• Proposition 3
• proof
• Definition 4: Compatibility graph
• Theorem 5
• proof
• Proposition 6
• proof
• Theorem 7
• proof