Bipartite Graphs Are Not Well-Quasi-Ordered by Bipartite Minors

In "Bipartite minors" [Journal of Combinatorial Theory, Series B, 2016], Chudnovsky et al. introduced the bipartite minor relation, a quasi-order on the class of bipartite graphs somewhat analogous the minor relation on general graphs and asked whether it is a well-quasi-order. We answer this question negatively by giving an infinite set of 2-connected bipartite graphs that are pairwise incomparable with respect to the bipartite minor relation. We additionally give two sets of infinitely many pairs of bipartite graphs: one set of pairs G, H such that H is a bipartite minor, but not a minor, of G, and one set of pairs G, H such that H is a minor, but not a bipartite minor, of G.