Universal-exponent diameter algorithms for minor-free graphs

Establish whether there exists a universal constant c > 0 such that, for every integer h > 1, the diameter problem for unweighted, undirected n-vertex K_h-minor-free graphs can be solved in time O_h(n^{2-c}).

Background

The paper studies subquadratic algorithms for computing graph diameter and all vertex eccentricities in graphs of bounded Euler genus and related graph classes. Existing algorithms for K_h-minor-free graphs achieve a running time of O_h(n{2-c_h}), where the improvement c_h decreases with h. The authors note that known conditional lower bounds do not rule out a universal exponent improvement independent of h.

The conjecture is verified for graphs of bounded Euler genus, graphs of bounded Euler genus with a constant number of apices, and certain clique-sum decompositions. However, the full K_h-minor-free case remains unresolved because the graph-minor structure theorem also involves vortices, which are not handled by the paper's techniques.

References

However, known lower bounds, including the one of, does not exclude the possibility that $c_h$ can be made a universal constant. That is, no known lower bound refutes the following conjecture: There exists a constant $c > 0$ such that, for every integer $h > 1$, the diameter problem in (unweighted, undirected) $n$-vertex $K_h$-minor-free graphs can be solved in time $O_h(n{2-c})$.

— Faster diameter computation in graphs of bounded Euler genus  (2502.07501 - Kluk et al., 11 Feb 2025) in Introduction, immediately following Conjecture 1 (labelled conjecture:taunt)