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

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 lette...

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Bibliographic Details
Published in:	Algorithmica
Main Authors:	Alecu, Bogdan, Kanté, Mamadou Moustapha, Lozin, Vadim, Zamaraev, Viktor
Format:	Journal Article
Language:	English
Published:	Springer Verlag 19.02.2024
Subjects:	Computational Complexity Computer Science Data Structures and Algorithms Discrete Mathematics Discrete Mathematics (cs.DM) FOS: Computer and information sciences FOS: Mathematics Data Structures and Algorithms (cs.DS) Combinatorics (math.CO)
ISSN:	0178-4617, 1432-0541
Online Access:	Get full text
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Summary:	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.
ISSN:	0178-4617 1432-0541
DOI:	10.48550/arXiv.2402.12559