In this paper, we investigate three fundamental problems regarding cut complexes of graphs: their realizability, the uniqueness of graph reconstruction from them, and their algorithmic recognition. We define the parameter m(d,n) as the minimum number of additional vertices needed to realize any d-dimensional simplicial complex on n vertices as a cut complex, and prove foundational bounds. Furthermore, we characterize precisely when a graph on n≥5 vertices is uniquely reconstructible from its 3-cut complex. Based on this characterization, we develop an O(n4) recognition algorithm. These results deepen the connection between graph structure and the topology of cut complexes.
@article{arxiv.2512.12933,
title = {Realizations and Uniqueness of Cut Complexes of Graphs},
author = {Yufeng Shen and Zhiyu Song and Fenglin Yu and Leopold Wuhan Zhou and Jingqi Zhuang},
journal= {arXiv preprint arXiv:2512.12933},
year = {2025}
}