Planar cluster-count identity ($k({\bar{\omega }}^{\ast })=o(\omega )-|V|+k(\omega )+1$)

For a fixed finite plane graph, the complementary dual configuration opens a dual edge exactly when its primal edge is closed. Its components correspond to faces of the spanning open primal graph. The componentwise Euler formula for a connected planar graph gives the displayed identity. The outer face and isolated primal vertices are included; the identity also handles disconnected graphs, bridges and loops.