Realizations and Uniqueness of Cut Complexes of Graphs

TL;DR

This paper explores the realizability, uniqueness, and recognition of cut complexes of graphs, establishing bounds, characterizations, and algorithms to understand their structure and reconstructibility.

Contribution

It introduces foundational bounds for realizing cut complexes, characterizes when graphs are uniquely reconstructible from their 3-cut complexes, and develops an efficient recognition algorithm.

Findings

Established bounds for realization of cut complexes

Characterized unique reconstructibility for graphs with ≥5 vertices

Developed an O(n^4) recognition algorithm

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.