Paper
Realizations and Uniqueness of Cut Complexes of Graphs
Authors
Yufeng Shen, Zhiyu Song, Fenglin Yu, Leopold Wuhan Zhou, Jingqi Zhuang
Abstract
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 as the minimum number of additional vertices needed to realize any -dimensional simplicial complex on vertices as a cut complex, and prove foundational bounds. Furthermore, we characterize precisely when a graph on vertices is uniquely reconstructible from its -cut complex. Based on this characterization, we develop an recognition algorithm. These results deepen the connection between graph structure and the topology of cut complexes.