Behavior of graph families with unbounded maximum degree and sublinear minimum degree

Determine whether graph families with unbounded maximum degree and sublinear minimum degree can exhibit behavior different from the exponential growth in the number of forts and compatible collections established for trees, bounded-degree graphs, and graphs with linear minimum degree.

Background

The paper studies forts, which obstruct zero forcing, and compatible collections of forts relevant to maximum-nullity and minimum-rank bounds. It proves that the number of forts, and consequently the number of compatible collections, grows exponentially for every tree, every connected non-star graph of bounded maximum degree, and every connected graph whose minimum degree is linear in the order of the graph.

The authors leave unresolved whether graph families outside these classes—specifically, families with unbounded maximum degree and sublinear minimum degree—might have different asymptotic behavior, such as only polynomially many compatible collections of forts.