On $m$-Closed Graphs

A graph is closed when its vertices have a labeling by $[n]$ such that the binomial edge ideal $J_G$ has a quadratic Gr"{o}bner basis with respect to the lexicographic order induced by $x_1>\cdots >x_n>y_1>\cdots >y_n$. In this paper, we generalize this notion and study the so called $m-$closed graphs. We find equivalent condition to $3-$closed property of an arbitrary tree $T$. Using it, we classify a class of $3-$closed trees. The primary decomposition of this class of graphs is also studied.