Van Kampen diagrams, boundary labels, and diagram area for a presentation

Fix a presentation $G=⟨X\mid R⟩$ (Group presentation by generators and relations). A van Kampen diagram over this presentation is either:

• the degenerate one-vertex diagram, whose boundary label is the empty word; or
• a finite planar combinatorial 2-complex whose underlying space is a closed disc, whose oriented edges are labelled by letters of $X^{\pm 1}$, and whose every 2-cell has boundary word a cyclic conjugate of some relator in $R$ or of its inverse.

In the nondegenerate case, after choosing an orientation of the boundary circuit of the disc, the boundary label of the diagram is the word read along that circuit. The area of the diagram is the number of 2-cells it contains; this is a natural number because the 2-cell set is finite (The cardinality $\mid A\mid$ of a finite set).