Is the k-gap-planar crossing ratio 1-jumping?

Determine whether the k-gap-planar crossing ratio is 1-jumping; specifically, decide whether the worst-case ratio between the k-gap-planar crossing number and the (k+1)-gap-planar crossing number attains the same asymptotic order as the k-gap-planar crossing ratio against the unrestricted crossing number. Equivalently, ascertain whether the largest attainable ratio is already achieved when comparing parameterizations k and k+1 for k-gap-planar drawings.

Background

The paper introduces the notion of 1-jumping for parameterized beyond-planarity concepts: a crossing ratio is called 1-jumping if the worst-case asymptotic ratio is already achieved when the parameter k is increased by one (i.e., comparing k vs. k+1).

Using their framework, the authors establish 1-jumping for all considered parameterized concepts except k-gap-planarity. For k-gap-planarity, their constructions do not yield 1-jumping, leaving open whether this reflects a genuine property of the concept or merely a limitation of their method.

References

Interestingly, for every but one considered beyond-planarity concept \bp, we were able to show that the ratio is 1-jumping, i.e., we can observe the worst-case crossing ratio already for graphs that would attain the optimal crossing number if we were to increase $k$ by just~1. Only for the $k$-gap-planar crossing ratio, our construction does not yield 1-jumping; it is unclear whether this is a limitation of our framework or if the $k$-gap-planar crossing ratio is indeed not 1-jumping.

— Crossing Numbers of Beyond Planar Graphs Re-revisited: A Framework Approach  (2407.05057 - Chimani et al., 2024) in Conclusion