S^1-flows after deleting one edge from a Z_3-connected graph

Determine whether, apart from a finite collection of small exceptional graphs, every graph H obtained by deleting one edge from a Z_3-connected graph has the property that whenever a graph G obtained by contracting H admits a nowhere-zero 3-flow, the graph G admits an S^1-flow.

Background

The paper proves that contracting a wheel subgraph preserves the existence of an S1-flow under the condition that the contracted graph admits a nowhere-zero 3-flow. It then asks whether the wheel can be replaced by a broader class of graphs.

The proposed class consists of graphs obtained by deleting one edge from a Z_3-connected graph. The authors point out that odd wheels are not Z_3-connected but become Z_3-connected after adding any missing edge, suggesting the relevance of this class, while the example 2K_2 indicates that some exceptions are unavoidable. The question asks whether only finitely many small exceptional graphs obstruct the analogous contraction property.

References

Another natural question is whether the wheel in Theorem~\ref{thm:intro-wheel-contraction-general} can be replaced by a broader class of graphs. An odd wheel is not $\mathbb{Z}_3$-connected, but adding any missing edge makes it $\mathbb{Z}_3$-connected. This suggests that a similar result may hold for graphs obtained by deleting one edge from a $\mathbb{Z}_3$-connected graph. However, $2K_2$ shows that some exceptions are unavoidable.

Apart from a finite collection of small exceptional graphs, does every graph $H$ obtained by deleting one edge from a $\mathbb{Z}_3$-connected graph have the following property: if $G/H$ admits a nowhere-zero $3$-flow, then $G$ admits an $S1$-flow?

— Reduction Operations and Structural Characterizations of $S^1$-Flows in Graphs  (2608.18725 - Li et al., 19 Aug 2026) in Section 5.2, “Future Directions on Vector Flows and Graph Reductions”