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arXiv · 2512.12933

Realizations and Uniqueness of Cut Complexes of Graphs

Yufeng Shen·Zhiyu Song·Fenglin Yu·Leopold Wuhan Zhou·Jingqi Zhuang
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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 $m(d,n)$ 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 $n \geq 5$ vertices is uniquely reconstructible from its $3$-cut complex. Based on this characterization, we develop an $O(n^4)$ recognition algorithm. These results deepen the connection between graph structure and the topology of cut complexes.

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Yufeng Shen, Zhiyu Song, Fenglin Yu, Leopold Wuhan Zhou, Jingqi Zhuang. 2025-12-15. Realizations and Uniqueness of Cut Complexes of Graphs. https://arxiv.org/abs/2512.12933

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