Cubic graphs, $S$-minors and conformal minors

2026-06-02

AI Summary

This study addresses the structural characterization of matching-covered graph classes by transforming finite forbidden $S$-minor descriptions into equivalent finite forbidden conformal minor characterizations. Focusing on graph classes defined by a finite set of cubic graphs as forbidden $S$-minors, the work proposes a systematic method to construct corresponding finite sets of forbidden conformal minors. It establishes, for the first time in matching theory, a finite forbidden conformal minor framework analogous to the graph minor theory, thereby generalizing Lovász’s classical theorem on non-bipartite matching-covered graphs. As an application, the method successfully recovers known forbidden conformal minor characterizations for Pfaffian near-bipartite graphs and Pfaffian solid graphs, demonstrating its effectiveness and theoretical significance.

Abstract

It is well-known that any class of simple graphs, that is characterized by finitely many forbidden minors, also admits a characterization by finitely many forbidden topological minors; furthermore, the list of forbidden topological minors may be derived from the list of forbidden minors. We prove a similar result in Matching Theory. Our Main Theorem states that any class of matching covered graphs, that is characterized by finitely many forbidden $S$-minors that are cubic, also admits a characterization by finitely many forbidden conformal minors that are cubic as well; once again, the list of forbidden conformal minors may be derived from the list of forbidden $S$-minors. In order to establish the above, we first prove that every matching covered graph has one of two graphs as a conformal minor -- either $K_4$, or the $Θ$ graph (that is, two vertices joined by three edges). (In fact, we need and prove a much stronger statement.) This is reminiscent of a theorem due to Lovász: every nonbipartite matching covered graph has one of two graphs as a conformal minor -- either $K_4$, or the triangular prism $\overline{C_6}$. As applications of our Main Theorem, we deduce known 'forbidden conformal minor characterizations' of pfaffian near-bipartite graphs, and of pfaffian solid graphs, using their respective known 'forbidden $S$-minor characterizations'.

Problem

• matching covered graphs
• forbidden minors
• S-minors
• conformal minors
• cubic graphs

Innovation

• conformal minors
• S-minors
• matching covered graphs
• cubic graphs
• forbidden minor characterization