Geodesic
Rees algebras, vertex covers and irreducible representations of Rees cones
Algebra and discrete mathematics, Tome 10 (2010) no. 2, pp. 64-86.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $G$.
Keywords: edge ideal, symbolic Rees algebras, perfect graph, irreducible vertex covers, irreducible graph, Alexander dual, blocker
Mots-clés : clutter. 
@article{ADM_2010_10_2_a5,
     author = {L. A. Dupont and R. N. Villarreal},
     title = {Rees algebras, vertex covers and irreducible representations of {Rees} cones},
     journal = {Algebra and discrete mathematics},
     pages = {64--86},
     publisher = {mathdoc},
     volume = {10},
     number = {2},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2010_10_2_a5/}
}
TY  - JOUR
AU  - L. A. Dupont
AU  - R. N. Villarreal
TI  - Rees algebras, vertex covers and irreducible representations of Rees cones
JO  - Algebra and discrete mathematics
PY  - 2010
SP  - 64
EP  - 86
VL  - 10
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2010_10_2_a5/
LA  - en
ID  - ADM_2010_10_2_a5
ER  - 
%0 Journal Article
%A L. A. Dupont
%A R. N. Villarreal
%T Rees algebras, vertex covers and irreducible representations of Rees cones
%J Algebra and discrete mathematics
%D 2010
%P 64-86
%V 10
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2010_10_2_a5/
%G en
%F ADM_2010_10_2_a5
L. A. Dupont; R. N. Villarreal. Rees algebras, vertex covers and irreducible representations of Rees cones. Algebra and discrete mathematics, Tome 10 (2010) no. 2, pp. 64-86. http://geodesic.mathdoc.fr/item/ADM_2010_10_2_a5/