Efficient computation of displacement

Determine whether there exist efficient algorithms for computing the displacement _r(G) of the r-local covering of a graph G.

Background

The displacement _r(G) is defined using the r-local covering p_r:G_rG, which is usually infinite. It serves as a graph invariant controlling the range of orders k for which the finite combinatorial theory of r-local separations and bottlenecks is guaranteed to work: specifically, the guarantee applies when k<K(G,r), where K(G,r)=_r(G)/r+1. Although the paper supplies finite combinatorial lower bounds for _r(G), the authors explicitly leave unresolved whether the displacement itself can be determined efficiently.

References

The definition of the displacement _r(G) involves the r-local covering, which is usually infinite, and so it is not clear whether there exist efficient algorithms that determine _r(G).

— Canonical graph decompositions and local separations: From infinite coverings to a finite combinatorial theory  (2501.16170 - Carmesin et al., 27 Jan 2025) in Section 1, subsection “Graph-decompositions and local separations”