Multigraphs without large bonds are wqo by contraction

TL;DR

This paper proves that multigraphs with limited connected components and small bonds are well-quasi-ordered under edge contraction, providing a structural understanding and canonical antichains for this class.

Contribution

It establishes that such multigraphs are well-quasi-ordered by edge contraction and characterizes their canonical antichains, advancing graph minor theory.

Findings

Multigraphs with bounded components and bonds are wqo by contraction.

Canonical antichains for this class are characterized and shown to be fundamental.

The results extend understanding of graph minors and contractions.

Abstract

We show that the class of multigraphs with at most $p$ connected components and bonds of size at most $k$ is well-quasi-ordered by edge contraction for all positive integers $p,k$. (A bond is a minimal non-empty edge cut.) We also characterize canonical antichains for this relation and show that they are fundamental.