Effect of graph combinations on largest neighbor parking functions

Determine how different operations for combining graphs, including gluing along subgraphs and taking graph products, affect the largest neighbor parking functions of the resulting graphs.

Background

The paper defines largest neighbor parking functions on labeled graphs and proves that for a disjoint union of graphs, the associated parking functions are obtained by shuffling largest neighbor parking functions from the connected components. It notes that other graph-combination operations, such as gluing along subgraphs and taking graph products, need not be governed by this shuffle description, and explicitly asks how these operations affect the resulting parking-function families.

References

How do different ways of combining graphs affect their largest neighbor parking functions?

— Parking with Frustrated Drivers  (2609.35638 - Hallam et al., 28 Sep 2026) in Section 'Open Questions', Question following Proposition \ref{prop:neighborShuffle}