Joins and ear decompositions beyond graphic matroids

2026-08-02

AI Summary

This study investigates whether the equality in Frank’s theorem—relating the maximum join size μ(M) and the ear-decomposition parameter η(M)—holds for general matroids. Leveraging matroid theory, NP-hardness reductions, inapproximability analysis, and Seymour’s decomposition theorem for regular matroids, the authors establish that this equality fails for both co-graphic and sparse paving matroids, and moreover, these classes admit no forbidden-minor characterization for the property. They further prove that computing the maximum join is NP-hard and inapproximable within any constant factor for co-graphic matroids. Finally, they derive a universal upper bound for regular matroids, showing that η(M) ≤ 6μ(M) − 2.

Abstract

For a matroid $M$, a join is a set $J\subseteq E(M)$ that meets every circuit $C$ in at most $|C|/2$ elements. Let $μ(M)$ denote the maximum size of a join. Motivated by Frank's min--max theorem for graphic matroids, we compare $μ(M)$ with an ear-decomposition parameter $η(M)=(r(M)+\varphi(M))/2$, where $\varphi(M)$ is the minimum number of even lobes in an ear decomposition of $M$. Frank's theorem implies $μ(M)=η(M)$ for connected graphic matroids. Here we study how far this equality extends beyond graphic matroids. We show that the exact equality does not hold in general: it already fails for cographic matroids, hence within the binary class. Furthermore, the class of matroids satisfying $μ(M)=η(M)$ is not minor-closed, thus there is little hope for a forbidden minor characterization. We also prove that computing a maximum join is NP-hard for cographic matroids, hard to approximate within a factor of $519/520$, and NP-hard for sparse paving matroids given by their list of bases. Despite these negative results, we show that the two parameters remain quantitatively comparable in several natural classes. We prove comparison bounds for binary, paving, cographic, and arbitrary connected matroids. In particular, using Seymour's decomposition theorem, we combine the equality for graphic matroids, the bound for cographic matroids, and a direct analysis of $R_{10}$ to obtain $η(M)\leq 6μ(M)-2$ for every regular matroid $M$.

Problem

• matroid
• join
• ear decomposition
• graphic matroid
• cographic matroid

Innovation

• matroid
• join
• ear decomposition
• regular matroid
• NP-hardness