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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600016X&_mathId=si1.gif&_user=111111111&_pii=S074771711600016X&_rdoc=1&_issn=07477171&md5=d2d94047a2208efde33418b0e2c02d27" title="Click to view the MathML source">∀θ∃Δclass="mathContainer hidden">class="mathCode">$\forall \theta \exists \mathrm{\Delta }$-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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600016X&_mathId=si2.gif&_user=111111111&_pii=S074771711600016X&_rdoc=1&_issn=07477171&md5=497b6a24c660b14dc4bf176b3f8f74b1" title="Click to view the MathML source">CIOclass="mathContainer hidden">class="mathCode">$\mathrm{CI}\mathcal{O}$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600016X&_mathId=si294.gif&_user=111111111&_pii=S074771711600016X&_rdoc=1&_issn=07477171&md5=076d2419342df5c3a81a869188df0ee4" title="Click to view the MathML source">K₄class="mathContainer hidden">class="mathCode">$K_4$. We introduce a new graph invariant, the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600016X&_mathId=si2.gif&_user=111111111&_pii=S074771711600016X&_rdoc=1&_issn=07477171&md5=497b6a24c660b14dc4bf176b3f8f74b1" title="Click to view the MathML source">CIOclass="mathContainer hidden">class="mathCode">$\mathrm{CI}\mathcal{O}$ deficiency. This invariant has the property that graphs with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600016X&_mathId=si2.gif&_user=111111111&_pii=S074771711600016X&_rdoc=1&_issn=07477171&md5=497b6a24c660b14dc4bf176b3f8f74b1" title="Click to view the MathML source">CIOclass="mathContainer hidden">class="mathCode">$\mathrm{CI}\mathcal{O}$ deficiency zero are exactly class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600016X&_mathId=si2.gif&_user=111111111&_pii=S074771711600016X&_rdoc=1&_issn=07477171&md5=497b6a24c660b14dc4bf176b3f8f74b1" title="Click to view the MathML source">CIOclass="mathContainer hidden">class="mathCode">$\mathrm{CI}\mathcal{O}$ graphs.