The Grobner Basis of Binomial Edge Ideals of Some Class of Graphs

A closed graph is a graph whose binomial edge ideal has a quadratic Grobner basis with respect to a suitable ordering of its vertices. This class of graphs is studied well in the literatures. We extend this notion and study the class of so called m−closed graphs. We present an equivalent condition for 3-closed property of trees. In particular we study caterpillar trees and some trees constructed by this class of graphs.