The generation of footpaths for additive manufacturing (AM), a process commonly known as “slicing,” has a strong impact on the performance of both the associated hardware systems and the resulting objects. Available slicers invariably produce discontinuous tootpaths, featuring jumps or so-catted “travel moves” during which the deposition of material or/and energy must be hatted. For AM processes using slowly solidifying feedstock materials, such as thermosetting polymers or cementitious mixtures such as concrete, these tootpath discontinuities are highly undesirable due to the artifacts they generate. This renders existing sticers difficult to use in such applications, and presents a road-block to the adoption of AM for such material systems. In the present work, this difficulty is addressed by the development of a simple geometric criterion for the existence of continuous tool-paths that are capable of producing a specified input geometry. This development is based on the principles of morphological geometric analysis and graph theory. It is shown that, for any geometric feature with a characteristic thickness at least twice the extrusion width, a continuous toolpath exists. Furthermore, a general-purpose algorithm for continuous toolpath generation, for arbitrarily shaped objects satisfying this criterion, is developed and demonstrated on a representative test problem. Finally, conclusions and the path forward for the usage of this approach with existing AM systems is explored.