  Herzog-Hibi-Hreind\'{o}ttir-Kahle-Rauh introduced the class of closed graph
and they proved that the binomial edge ideal $J(G)$ of a graph $G$ has
quadratic Gr\"{o}bner bases if $G$ is closed. In this paper, we introduce the
class of weakly closed graph as a generalization of the closed graph and prove
that the quotient ring $S/J(G)$ is $F$-pure if $G$ is weakly closed. This fact
is a generalization of Ohtani's theorem.

Commutative Algebra;Combinatorics