A <disc diagram> is reduced if it has no adjacent pair of two-cells that can be folded together across their common boundary and removed while preserving the outside boundary. A minimum-area diagram is reduced. Among diagrams of minimum area, minimizing the number of edges removes unnecessary tree portions. Internal common arcs in a reduced diagram satisfy the applicable <small cancellation theory> piece bounds.


Reduced disc diagram

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Small cancellation theory controls overlap between <relators> or attaching paths of polygonal two-cells. Requiring overlaps to be small compared with cell perimeters gives strong <disc diagram> structure, shortening algorithms and geometric restrictions on subgroups. The <metric small cancellation condition> $C'(1/6)$ is a central example.


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