= Planar cluster-count identity
{title2=$k(\bar\omega^*)=o(\omega)-|V|+k(\omega)+1$}
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.
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