CIO and ring graphs: Deficiency and testing

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• 作者：Isidoro Gitler ^{igitler@math.cinvestav.edu.mx" class="auth_mail" title="E-mail the corresponding author} ; Enrique Reyes ^{ereyes@math.cinvestav.mx" class="auth_mail" title="E-mail the corresponding author} ; Juan A. Vega ^{javega@math.cinvestav.edu.mx" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Complete intersection toric ideals ; Oriented graphs ; Prisms ; Pyramids ; Thetas ; Ring graphs ; Chorded-thetas
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：March-April 2017
• 年：2017
• 卷：79
• 期：part_P2
• 页码：249-268
• 全文大小：759 K

文摘

We give a polynomial time algorithm that tests whether a graph is a theta-ring graph or equivalently if a graph has the ∀θ∃Δ$\forall \theta \exists \mathrm{\Delta }$-property (each chorded-theta has a transversal triangle). It is known that G is a theta-ring graph if and only if G   is a CIO$\mathrm{CI}\mathcal{O}$ graph (each toric ideal associated to an edge orientation of G   is a binomial complete intersection). In particular ring graphs are theta-ring graphs. We prove that the forbidden induced subgraphs that characterize ring graphs are chorded-thetas and K₄$K_4$. We introduce a new graph invariant, the CIO$\mathrm{CI}\mathcal{O}$ deficiency. This invariant has the property that graphs with CIO$\mathrm{CI}\mathcal{O}$ deficiency zero are exactly CIO$\mathrm{CI}\mathcal{O}$ graphs.