Bipartite Induced Subgraphs and Well-Quasi-Ordering

TL;DR

This paper investigates the well-quasi-ordering of bipartite graphs under the induced subgraph relation, establishing that $P_6$-free bipartite graphs are wqo while $P_7$-free are not, advancing understanding of graph classes.

Contribution

It resolves an open question by proving $P_7$-free bipartite graphs are not wqo and shows $P_6$-free bipartite graphs are wqo, with partial results on more restricted subclasses.

Findings

$P_7$-free bipartite graphs are not wqo.

$P_6$-free bipartite graphs are wqo.

Partial results on subclasses with multiple forbidden induced subgraphs.

Abstract

We study bipartite graphs partially ordered by the induced subgraph relation. Our goal is to distinguish classes of bipartite graphs which are or are not well-quasi-ordered (wqo) by this relation. Answering an open question from \cite{Ding92}, we prove that $P_7$-free bipartite graphs are not wqo. On the other hand, we show that $P_6$-free bipartite graphs are wqo. We also obtain some partial results on subclasses of bipartite graphs defined by forbidding more than one induced subgraph.