Subquadratic recognition of bounded-genus graphs with apices and their decompositions

Construct, in subquadratic time, a set of at most k apex vertices whose deletion leaves a graph of Euler genus at most g, or construct the corresponding tree decomposition with bounded-size apex sets and bounded-size adhesions, even approximately, so that the input requirements in the bounded-genus-with-apices and clique-sum diameter algorithms can be removed.

Background

The algorithmic results for graphs of bounded Euler genus with apices and for clique-sums of such graphs assume that the relevant apex set or tree decomposition is supplied as part of the input. Thus, the results are conditional on having these structural objects available.

The authors explicitly state that they are unaware of an existing subquadratic algorithm that finds the required apex set or decomposition, even approximately. They mention that recent almost-linear-time minor-testing techniques may potentially yield such a construction, but leave the task unresolved.

References

At this point, we are not aware of any existing algorithm that would find in subquadratic time a set $A$ as in \Cref{thm:main-apices}, or the decomposition $(T,\beta)$ with the sets $A$ as in \Cref{thm:main-decomp}, even in the approximate sense. However, we were informed by Korhonen, Pilipczuk, Stamoulis, and Thilikos that it seems likely that the techniques introduced in the recent almost linear-time algorithm for minor-testing could be used to construct such an algorithm, with almost linear time complexity. With this result in place, the assumption about the decomposition and/or apex sets being provided on input could be lifted in \Cref{thm:main-apices,thm:main-decomp}; this is, however, left to future work.

— Faster diameter computation in graphs of bounded Euler genus  (2502.07501 - Kluk et al., 11 Feb 2025) in Section 1, paragraph "Generalizations", immediately after Theorem 1.6