Cell neighbor exchanges play a critical role in regulating tissue fluidity during epithelial morphogenesis and repair. In vivo, these neighbor exchanges are often hindered by the formation of transiently stable fourfold vertices, which can develop into complex multicellular rosettes where five or more cell junctions meet. Despite their importance, the mechanical origins of multicellular rosettes have remained elusive, and current cellular models lack the ability to explain their formation and maintenance. Here we present a dynamic vertex model of epithelial tissues with strain-dependent tension remodeling and mechanical memory dissipation. We show that an increase in cell junction tension upon contraction and reduction in tension upon extension can stabilize higher-order vertices, temporarily stalling cell rearrangements. On the other hand, inducing mechanical memory dissipation via relaxation of junction strain and stress promotes the resolution of higher-order vertices, facilitating cell neighbor exchanges. We demonstrate that by tuning the rates of tension remodeling and mechanical memory dissipation, we can control topological transitions and tissue material properties, recapitulating complex cellular topologies seen in developing organisms.