Coté, B.; Ewing, C.; Huhn, M.; Plaut, C. M.; Weber, D. Cut-sets in zero-divisor graphs of finite commutative rings. (English) Zbl 1228.13011 Commun. Algebra 39, No. 8, 2849-2861 (2011). Let \(R\) be a commutative ring with nonzero identity, and let \(Z(R)\) be its set of zero-divisors. The zero-divisor graph, \(\Gamma(R)\), is the graph with vertices the set of nonzero zero-divisors of \(R\), and for distinct \(x,y \in {Z(R)}\), the vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\). The zero-divisor graph of a commutative ring has been studied extensively by several authors. For an arbitrary connected graph \(G\) with the set of vertices \(V(G)\), a set \(A\subset V(G)\) is said to be a cut-set if there exist distinct \(c, d\in V(G)\setminus A\) such that every path in \(G\) from \(c\) to \(d\) involves at least one element of \(A\), and no proper subset of \(A\) satisfies the same condition. In the paper under review the authors examine minimal sets of vertices which, when removed from a zero-divisor graph, separate the graph into disconnected subgraphs. In addition, the authors classify these sets for all finite nonlocal commutative rings with identity. Reviewer: Siamak Yassemi (Tehran) Cited in 3 Documents MSC: 13A99 General commutative ring theory Keywords:zero-divisor graph; cut-set Software:Mathematica PDFBibTeX XMLCite \textit{B. Coté} et al., Commun. Algebra 39, No. 8, 2849--2861 (2011; Zbl 1228.13011) Full Text: DOI References: [1] Anderson D. F., J. Algebra 217 pp 434– (1999) · Zbl 0941.05062 · doi:10.1006/jabr.1998.7840 [2] Axtell M., Involve 2 pp 17– (2009) · Zbl 1169.13301 · doi:10.2140/involve.2009.2.17 [3] Dummit D. S., Abstract Algebra., 3. ed. (2004) [4] Redmond S., Discrete Math. 307 pp 1155– (2007) · Zbl 1107.13006 · doi:10.1016/j.disc.2006.07.025 [5] MATHEMATICA ( 2008 ). Wolfram Research, Inc., Mathematica, Version 7.0, Champaign, IL . This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.