Vertex-based equivalence between triangle listing and all-edge detection

Determine whether listing n^2 triangles is equivalent to all-edge triangle detection when runtimes are measured as functions of the number of vertices n.

Background

Known reductions establish an equivalence between all-edge triangle detection and listing m triangles when runtimes are expressed in terms of the number of edges m. The paper asks whether an analogous equivalence holds for dense, vertex-parameterized problems, namely listing n2 triangles and all-edge detection.

One direction is obtained: all-edge triangle detection reduces to listing n2 triangles for the relevant range of exponents. The reverse direction remains unresolved. The paper gives only a quantitatively weaker reduction under the assumption that integer matrix multiplication can be performed in quadratic time and proves barriers for non-duplicitous black-box reductions.

References

However, in the other direction, such a translation is unclear.

— The Limits of Black-Box Reductions for All-Pairs Triangle Detection  (2608.19092 - Sheffield et al., 19 Aug 2026) in Section 1, subsection “Reducing listing to all-edge detection” (Question 3)

A strengthening to arbitrary black-box reductions may be possible, but we do not know how to prove it.

— The Limits of Black-Box Reductions for All-Pairs Triangle Detection  (2608.19092 - Sheffield et al., 19 Aug 2026) in Section 1, subsection “Listing to all-edge detection with non-duplicitous reductions”