Increase the gap between leaves in BFS first-in trees and general spanning trees

Determine whether there exists a family of graphs for which the maximum number of leaves over all Breadth-First Search (BFS) first-in spanning trees is bounded by a constant, while the same graphs admit general spanning trees with Ω(n) leaves, thereby increasing the known gap beyond the current Ω(√n) vs. Ω(n) separation.

Background

The paper contrasts the number of leaves achievable in search trees versus unrestricted spanning trees. For minimization, they exhibit graphs where the minimum number of leaves in a BFS first-in tree can be linear in n even though a spanning tree with a single leaf exists. For maximization, they present a family (a star of ladders) where any BFS first-in tree has O(√n) leaves, while some spanning trees have Ω(n) leaves.

They explicitly ask whether this separation can be pushed further, specifically whether one can find graphs whose BFS first-in trees all have only a constant number of leaves, while the graphs still admit spanning trees with a linear number of leaves.

References

We leave open whether this gap can be increased, i.e., whether there is a family of graphs in which the maximum number of leaves in any -tree is constant and there are spanning trees with Ω(n) leaves.

— Breadth-First Search Trees with Many or Few Leaves  (2604.00691 - Beisegel et al., 1 Apr 2026) in Section 3 (Number of Leaves as Parameter), paragraph following Figure 1