  Homogeneous Yang-Baxter (YB) deformation of AdS$_5\times$S$^5$ superstring is
revisited. We calculate the YB sigma model action up to quadratic order in
fermions and show that homogeneous YB deformations are equivalent to
$\beta$-deformations of the AdS$_5\times$S$^5$ background when the classical
$r$-matrices consist of bosonic generators. In order to make our discussion
clearer, we discuss YB deformations in terms of the double-vielbein formalism
of double field theory. We further provide an O(10,10)-invariant string action
that reproduces the Green-Schwarz type II superstring action up to quadratic
order in fermions. When an AdS background contains a non-vanishing $H$-flux, it
is not straightforward to perform homogeneous YB deformations. In order to get
any hint for such YB deformations, we study $\beta$-deformations of $H$-fluxed
AdS backgrounds and obtain various solutions of (generalized) type II
supergravity.
