Azuma et al. showed that a strongly connected digraph without a doubly bidirectionally connected pair is cactus-expandable. We prove the converse: if a digraph has a doubly bidirectionally connected pair, then no expansion of it is cactus.
@article{arxiv.2312.17327,
title = {Expansion of a digraph with a doubly bidirectionally connected pair is not cactus},
author = {Hiroki Kodama},
journal= {arXiv preprint arXiv:2312.17327},
year = {2026}
}