Zero-divisor graphs, and more recently, compressed zero-divisor graphs are well represented in the commutative ring literature. In this work, we consider various cut structures, sets of edges or vertices whose removal disconnects the graph, in both compressed and non-compressed zero-divisor graphs. In doing so, we connect these graph-theoretic concepts with algebraic notions and provide realization theorems of zero-divisor graphs for commutative rings with identity.
"Cut structures in zero-divisor graphs of commutative rings." J. Commut. Algebra 8 (2) 143 - 171, 2016. https://doi.org/10.1216/JCA-2016-8-2-143