On matrix and lattice ideals of digraphs

Let G be a simpleý, ýoriented connected graph with n vertices and m edgesý. ýLet I(B) be the binomial ideal associated to the incidence matrix \textbf{B} of the graph Gý. ýAssume that IL is the lattice ideal associated to the rows of the matrix Bý. ýAlso let Bi be a submatrix of B after removing the i-th rowý. ýWe introduce a graph theoretical criterion for G which is a sufficient and necessary condition for I(B)=I(Bi) and I(Bi)=ILý. ýAfter that we introduce another graph theoretical criterion for G which is a sufficient and necessary condition for I(B)=ILý. ýIt is shown that the heights of I(B) and I(Bi) are equal to n−1 and the dimensions of I(B) and I(Bi) are equal to m−n; then I(Bi) is a complete intersection idealý.