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Bipartite Graphs Are Not Well-Ordered by Bipartite Minors

Therese Biedl, Dinis Vitorino

TL;DR

This paper investigates whether the bipartite minor relation is a well-order on bipartite graphs. It uses explicit constructions to contrast the bipartite minor and ordinary minor relations, showing that both implications can fail by producing infinite families of counterexamples based on bulls $B(l,l_1)$ and dogs $D(l,l_1,l_2)$. It proves the presence of an infinite antichain $\{D(k,4,4)\mid k$ even $\ge 4\}$ among $2$-connected bipartite graphs, thereby ruling out a well-order for $\leq_B$ in this natural subfamily and suggesting a fundamental separation between the two containment notions. The results imply that forbidding bipartite minors does not suffice to capture all minor-closed properties of bipartite graphs and motivate further study of the limits of forbidden bipartite minors within minor-closed classes.

Abstract

In "Bipartite minors," Chudnovsky etal. introduced the bipartite minor relation, a partial order on the set of bipartite graphs somewhat analogous the minor relation on general graphs and asked whether it is a well-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 $G$.

Bipartite Graphs Are Not Well-Ordered by Bipartite Minors

TL;DR

This paper investigates whether the bipartite minor relation is a well-order on bipartite graphs. It uses explicit constructions to contrast the bipartite minor and ordinary minor relations, showing that both implications can fail by producing infinite families of counterexamples based on bulls and dogs . It proves the presence of an infinite antichain even among -connected bipartite graphs, thereby ruling out a well-order for in this natural subfamily and suggesting a fundamental separation between the two containment notions. The results imply that forbidding bipartite minors does not suffice to capture all minor-closed properties of bipartite graphs and motivate further study of the limits of forbidden bipartite minors within minor-closed classes.

Abstract

In "Bipartite minors," Chudnovsky etal. introduced the bipartite minor relation, a partial order on the set of bipartite graphs somewhat analogous the minor relation on general graphs and asked whether it is a well-order. We answer this question negatively by giving an infinite set of -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 such that is a bipartite minor, but not a minor, of , and one set of pairs such that is a minor, but not a bipartite minor, of .
Paper Structure (6 sections, 3 theorems, 1 equation, 6 figures)

This paper contains 6 sections, 3 theorems, 1 equation, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bipartite Minors vs. Minors
  4. Bipartite Minor, but Not Minor
  5. Minor, but Not Bipartite Minor
  6. Bipartite Minors and Well-Order

Key Result

Theorem 1

For all integers $l\geq 3$ and $l_1\geq 1$, the bull $B(l,l_1)$ is a bipartite minor of $C_{l+2l_1}$, but $B(l,l_1)$ is not a minor of $C_p$ for any integer $p\geq 3$.

Figures (6)

  • Figure 1: Some examples of bulls. Here and elsewhere, dashed lines indicate edges or induced paths.
  • Figure 2: Some examples of dogs.
  • Figure 3: $C_6$ and $B(4,1)$
  • Figure 4: The three smallest elements of a set of forests that are pairwise incomparable with respect to the subgraph relation
  • Figure 5: Contracting the cycle $C_8$ into the bull $B(4,2)$. The vertices that are contracted in each step are drawn in blue.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2: bipartiteMinors
  • Definition 3: bipartiteMinors
  • Theorem 1
  • proof
  • Theorem 2
  • proof : Proof of \ref{['thm: minor but not bip minor']}
  • Theorem 3
  • proof
  • Conjecture 1