The vertex model is highly successful in describing the motion of two-dimensional confluent cell tissues. In this model the energy of each cell is determined solely by its geometric shape and size, and the interaction between adjacent cells is a product of this additive energy. We expand the basic vertex model by considering open boundary conditions such that the tissue is allowed to be non-confluent. We identify several phases, two of which (solid confluent and liquid confluent) were found in previous studies that imposed confluency. Additionally, we prove that contrary to previous claims there cannot be a density-independent phase transition under periodic or hard-walls boundary conditions.