Relationship between McCarty-width parameters of consecutive orders

Determine the relationship between the order-f(2k) McCarty-width and order-f(2k+1) McCarty-width of a graph for every integer k2, beyond the inequalities currently established.

Background

The paper defines the McCarty-width of order k, denoted by _k(G), through balanced disk separators for every k-vertex subset. It proves that _{2k}(G) _{2k-1}(G), but gives examples showing that strict inequality can occur. The authors state that no general relationship is known between the even- and succeeding odd-order parameters _{2k}(G) and _{2k+1}(G).

References

We do not know the relationship between {2k}(G) and _{2k+1}(G), except that there are graphs G and integers k such that _{2k}(G)<{2k+1}(G).

— Graph parameters that are coarsely equivalent to tree-length  (2502.00951 - Dragan, 2 Feb 2025) in Section 3.2, subsection “McCarty-width of a graph”