Generalizing the graph-like domain construction beyond FGSL

Develop a general argument showing that, for any sufficiently smooth one-parameter family of metric graphs, one can construct corresponding families of smooth planar graph-like domains that (i) converge in C∞ to the graphs as the thickness parameter tends to zero and (ii) depend continuously and piecewise C1 on the parameter, thereby extending Proposition 4.1 beyond the specific geometry of families of planar graphs with a sliding loop (FGSL).

Background

Proposition 4.1 establishes, for a specific class of planar graphs (FGSL: smooth, locally straight families of planar graphs with a sliding loop), the existence of graph-like planar domains that converge to the graphs and have the required smooth parameter dependence.

The authors believe such a construction should hold for more general one-parameter families of metric graphs but lack an argument that avoids reliance on the special FGSL geometry.

References

We believe that this kind of result should hold for more general families of graphs. More precisely, we expect that for any one-parameter family of metric graphs depending in a sufficiently smooth way on the parameter, one should be able to contruct a family of domains satisfying the last two properties of the proposition. Yet as can be seen from the proof (especially Step 2) we were unable to find an argumentation that does not rely on the specific geometry of the FGSL.

— Neumann's nodal line may be closed on doubly-connected planar domains  (2604.03169 - Freitas et al., 3 Apr 2026) in Section “Graph-like domains with a sliding handle,” Remark following Proposition 4.1 (Proposition \ref{prop:graph-like domains})