  In this paper, we focus on contact structures supported by planar open book
decompositions. We study right-veering diffeomorphisms to keep track of
overtwistedness property of contact structures under some monodromy changes. As
an application we give infinitely many examples of overtwisted contact
structures supported by open books whose pages are the four-punctured sphere,
and also we prove that a certain family is holomorphically fillable using
lantern relation.
