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Edge polytopes is a class of interesting polytope with rich algebraic andcombinatorial properties, which was introduced by Ohsugi and Hibi. In thispapar, we follow a previous study on cutting edge polytopes by Hibi, Li andZhang. Instead of focusing on the algeraic properties of the subpolytopes asthe previous study, in this paper, we take a closer look on the graphs whoseedge polytopes are decomposable. In particular, we answer two importantquestions raised in the previous study about 1) the relationship between type Iand type II decomposable graphs and 2) description of decomposable graphs.

Decomposable edge polytopes of finite graphs