on matrix and lattice ideals of digraphs

‎let g be a simple‎, ‎oriented connected graph with n vertices and m edges‎. ‎let i(b) be the binomial ideal associated to the incidence matrix textbf{b} of the graph g‎. ‎assume that il is the lattice ideal associated to the rows of the matrix b‎. ‎also let bi be a submatrix of b after removing the i-th row‎. ‎we introduce a graph theoretical criterion for g which is a sufficient and necessary condition for i(b)=i(bi) and i(bi)=il‎. ‎after that we introduce another graph theoretical criterion for g which is a sufficient and necessary condition for i(b)=il‎. ‎it 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+1; then i(bi) is a complete intersection ideal‎.