Graphs without large bicliques and well-quasi-orderability by the induced subgraph relation

TL;DR

This paper proves that graphs excluding large bicliques are well-quasi-ordered by the induced subgraph relation and have bounded clique-width, enabling polynomial-time solutions for MSO-definable problems on such classes.

Contribution

It establishes that graphs without large bicliques are well-quasi-ordered by induced subgraphs and have bounded clique-width, confirming a conjecture for this class.

Findings

Graphs without large bicliques are well-quasi-ordered by induced subgraph relation.

Such graphs have bounded clique-width.

Algorithmic implications for solving MSO-definable problems in polynomial time.

Abstract

Recently, Daligault, Rao and Thomass\'e asked in [3] if every hereditary class which is well-quasi-ordered by the induced subgraph relation is of bounded clique-width. There are two reasons why this questions is interesting. First, it connects two seemingly unrelated notions. Second, if the question is answered affirmatively, this will have a strong algorithmic consequence. In particular, this will mean (through the use of Courcelle theorem [2]), that any problem definable in Monadic Second Order Logic can be solved in a polynomial time on any class well-quasi-ordered by the induced subgraph relation. In the present paper, we answer this question affirmatively for graphs without large bicliques. Thus the above algorithmic consequence is true, for example, for classes of graphs of bounded degree.