Cut ideals of \(K_{4}\)-minor free graphs are generated by quadrics
The cut ideal \(I_{G}\) of a finite graph \(G\) describe questions in applied science (statistics, optimization, computer science, etc) into questions in commutative algebra. In [\textit{B. Sturmfels} and \textit{S. Sullivant}, Mich. Math. J. 57, 689--709 (2008; Zbl 1180.13040)] the algebraic properties of cut ideals for graphs with up to six vertices were studied and a number of conjectures were outlined. The paper under review considers one of these conjectures, namely: The cut ideal \(I_{G}\) is generated by quadrics if and only if \(G\) is free of \(K_{4}\)-minors. The conjecture follows as a corollary of Theorem 2.6, which gives a description of generating set of \(I_G\), and was proven by induction on the number of vertices of \(G\).
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