Excluded conformal minors of Birkhoff-von Neumann graphs with equal global forcing number and maximum anti-forcing number

TL;DR

The paper investigates when the global forcing number $gf(G)$ equals the maximum anti-forcing number $Af(G)$ for matchable graphs, introducing the notion of strongly uniform graphs whose conformal subgraphs preserve this equality. It develops a structural theory built on ear decompositions and Hamilton cycles to identify excluded conformal minors, finding 29 critical conformal minors (4 from a small set $ m A$ and 25 from $ m D$) that prevent strong uniformity in BN-graphs. The authors delineate four BN-graph families $\ m \mathcal{G}_0$–$\mathcal{G}_3$ and prove: a BN-graph is strongly uniform iff it contains no conformal minor from $ m A\cup D$. They also verify, via construction and case analysis, that graphs in these families satisfy $gf(G)=Af(G)$, thereby providing a complete, minor-based dichotomy with potential implications for chemical graph theory and molecular stability metrics. The results combine combinatorial DP-tools with computational checks on fundamental BN-graphs to yield a comprehensive classification of strongly uniform conformal-minor-free BN-graphs.

Abstract

Global forcing number and maximum anti-forcing number of matchable graphs (graphs with a perfect matching) were proposed in completely different situations with applications in theoretical chemistry. Surprisingly for bipartite graphs and some nonbipartite graphs as solid bricks (or Birkhoff-von Neumann graphs) G, the global forcing number gf(G) is at least the maximum anti-forcing number Af(G). It is natural to consider when gf(G) = Af(G) holds. For convenience, we call a matchable graph G strongly uniform if each conformal matchable subgraph G' always satisfies gf(G') = Af(G'). In this article, by applying the ear decomposition theorem and discussing the existence of a Hamilton cycle with positions of chords, we give "excluded conformal minors" and "structural" characterizations of matchable bipartite graphs and Birkhoff-von Neumann graphs that are strongly uniform respectively.

Figures (17)

• Figure 1: A quadrilateral subdivision of $H_{1,4}$ on a strong replaceable set $\{v_1v_2,v_3v_4\}$.
• Figure 2: All graphs in $\mathcal{A}$.
• Figure 3: The forms of the graphs in $\mathcal{B}_0\cup \mathcal{B}_1\cup \mathcal{B}_2\cup \mathcal{B}_3$.
• Figure 4: Illustration for the proof of Claim \ref{['CP']}.
• Figure 5: Illustration for Claim \ref{['cla2']}.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Theorem 2.2
• Lemma 2.3
• Theorem 2.4
• Lemma 2.5
• Lemma 2.6
• proof