  In this work, we consider the full Horndeski Lagrangian applied to wormhole
geometries and present the full gravitational field equations. We analyse the
general constraints imposed by the flaring-out conditions at the wormhole
throat and consider a plethora of specific subclasses of the Horndeski
Lagrangian, namely, quintessence/phantom fields, $k$-essence, scalar-tensor
theories, covariant galileons, nonminimal kinetic coupling, kinetic gravity
braiding, and the scalar-tensor representation of Gauss-Bonnet couplings,
amongst others. The generic constraints analysed in this work serve as a
consistency check of the main solutions obtained in the literature and draws
out new avenues of research in considering applications of specific subclasses
of the Horndeski theory to wormhole physics.
