Let I = I(D) be the edge ideal of a weighted oriented graph D. We determine the irredundant irreducible decomposition of I. Also, we characterize the associated primes and the unmixed property of I. Furthermore, we give a combinatorial characterization for the unmixed property of I, when D is bipartite, D is a whisker or D is a cycle. Finally, we study the Cohen-Macaulay property of I. ⇐) If x ∈ L 3 (C), then by Proposition 2.4, N D (x) ⊆ C \ {x}. Hence, C \ {x} is a vertex cover. A contradiction, since C is minimal. Therefore L 3 (C) = ∅.Definition 2.6. A vertex cover C of D is strong if for each x ∈ L 3 (C) there is (y, x) ∈ E(D) such that y ∈ L 2 (C) ∪ L 3 (C) with y ∈ V + (i.e. w(y) = 1).

No abstract

Let [Formula: see text] be a connected simple graph, with [Formula: see text] vertices whose homogeneous monomial subring is [Formula: see text]. We prove that if [Formula: see text] is normal and Gorenstein, then [Formula: see text] is unmixed with cover number [Formula: see text] and [Formula: see text] has a strong [Formula: see text]-[Formula: see text]-reduction. Also, if [Formula: see text] is normal and [Formula: see text] is unmixed whose cover number is [Formula: see text], we give sufficient conditions for [Formula: see text] to be Gorenstein.