Tadpoles, Cephalopods, and `Complete Normal Ordering'

We describe how to cancel (when this is desirable) all tadpole and more generally all cephalopod Feynman diagrams in generic interacting scalar field theories to all orders in perturbation theory. This cancelation reduces the number of Feynman diagrams at a given loop order by a factor of two or more, and is accomplished by introducing the notion of 'complete normal ordering' (an extension of the standard field theory definition of normal ordering) which when applied to the bare action of the theories of interest results in tadpole- and cephalopod-free Greens functions.