Arc reduction and combinatorial curvature of a disc diagram

In a diagram of Sc toolkit labelled planar disc diagram, an arc is an edge path whose internal vertices have degree two and whose endpoints are vertices of other degrees. Replace each maximal such path by one edge labelled by its whole word; this is arc reduction. Edge lengths are retained as word lengths. Internal arcs have a face on both sides; exterior arcs border the unbounded region. A component that is a whole circle is retained with one marked vertex and one loop, rather than being suppressed to a vertex-free object. In particular use that convention for a one-face disc. A tree reduces to a tree with its spur tips retained. The isolated point is unchanged.

Give each face corner a real angle ${\alpha }_c$ measured in units of $\pi$. Let $d(f)$ count the edge occurrences around $f$. The link $\mathrm{lk}(v)$ is the finite graph with one vertex for each edge germ at $v$ and one edge for each incident face corner. Loop edges have two germs. Put $\chi (\mathrm{lk}(v))=V_{\mathrm{lk}(v)}-E_{\mathrm{lk}(v)}$ and

$k(f)={\sum }_{c\text{ at }f}{\alpha }_c-(d(f)-2),k(v)=2-\chi (\mathrm{lk}(v))-{\sum }_{c\text{ at }v}{\alpha }_c\mathrm{.}$

All incidences are counted with multiplicity. An interior disc vertex has circular link of Euler characteristic zero; an ordinary boundary vertex has interval link of Euler characteristic one. A spur tip has singleton link, no corners, and curvature one. The isolated point has empty link and curvature two. These angles are combinatorial data; they need not be geometrically realizable.