Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones

• Datum: 2018
• Hauptverfasser: Dupont, Luis A., Villarreal, Rafael H.
• Format: Artikel
• Sprache: English
• Veröffentlicht: Lugansk National Taras Shevchenko University 2018
• Schlagworte: edge ideal; symbolic Rees algebras; perfect graph; irreducible vertex covers; irreducible graph; Alexander dual; blocker; clutter; 13F20; 05C75; 05C65; 52B20
• Online Zugang: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/651
• Назва журналу: Algebra and Discrete Mathematics

Let \(G\) be a simple graph and let \(I_c(G)\) be its ideal of vertex covers. We give a graph theoretical description of the irreducible \(b\)-vertex covers of \(G\), i.e., we describe the minimal generators of the symbolic Rees algebra of \(I_c(G)\). Then we study the irreducible \(b\)-vertex covers of the blocker of \(G\), i.e., we study the minimal generators of the symbolic Rees algebra of the edge ideal of \(G\). We give a graph theoretical description of the irreducible binary \(b\)-vertex covers of the blocker of \(G\). It is shown that they correspond to irreducible induced subgraphs of \(G\). As a byproduct we obtain a method, using Hilbert bases, to obtain all irreducible induced subgraphs of \(G\). In particular we obtain all odd holes and antiholes. We study irreducible graphs and give a method to construct irreducible \(b\)-vertex covers of the blocker of \(G\) with high degree relative to the number of vertices of \(G\).