Abstract
Let $textit{G}$ be a simple, oriented connected graph with $n$ vertices and $m$ edges. Let $I(textbf{B})$ be the binomial ideal associated to the incidence matrix textbf{B} of the graph $G$. Assume that $I_L$ is the lattice ideal associated to the rows of the matrix $textbf{B}$. Also let $textbf{B}_i$ be a submatrix of $textbf{B}$ after removing the $i$-th row. We introduce a graph theoretical criterion for $G$ which is a sufficient and necessary condition for $I(textbf{B})=I(textbf{B}_i)$ and $I(textbf{B}_i)=I_L$. After that we introduce another graph theoretical criterion for $G$ which is a sufficient and necessary condition for $I(textbf{B})=I_L$. It is shown that the heights of $I(textbf{B})$ and $I(textbf{B}_i)$ are equal to $n-1$ and the dimensions of $I(textbf{B})$ and $I(textbf{B}_i)$ are equal to $m-n+1$; then $I(textbf{B}_i)$ is a complete intersection ideal.
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