  We consider a S=1 kagom\'e Ising model with triquadratic interactions around
each triangular face of the kagom\'e lattice, single-ion anisotropy and an
applied magnetic field. A mapping establishes an equivalence between the
magnetic canonical partition function of the model and the grand canonical
partition function of a kagom\'e lattice-gas model with localized
three-particle interactions. Since exact phase diagrams are known for
condensation in the one-parameter lattice-gas model, the mapping directly
provides the corresponding exact phase diagrams of the three-parameter S=1
Ising model. As anisotropy competes with interactions, results include the
appearance of confluent singularities effecting changes in the topology of the
phase diagrams, phase boundary curves (magnetic field vs temperature) with
purely positive or negative slopes as well as intermediate cases showing
nonmonotonicity, and coexistence curves (magnetization vs temperature) with
varying shapes and orientations, in some instances entrapping a homogeneous
phase.
