Realizations and Uniqueness of Cut Complexes of Graphs

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.

💡 Research Summary

The paper investigates three fundamental questions concerning cut complexes of finite simple graphs: (i) the realizability of arbitrary simplicial complexes as cut complexes, (ii) the uniqueness of graph reconstruction from a given cut complex, and (iii) the algorithmic recognition of such complexes.

A cut complex Δ_k(G) of a graph G on n vertices is defined as the simplicial complex whose facets are the complements of k‑subsets U⊆V(G) for which the induced subgraph G