Planar dual graph

A planar dual graph has one vertex in each face of a fixed planar embedding and one dual edge crossing each primal edge.

This is the planar-embedding case of a dual graph.

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.

In planar bond percolation, declare a dual edge open exactly when the primal edge it crosses is closed. Independent primal parameter $p$ then gives independent dual parameter $1-p$.

A planar finite network with three boundary terminals induces one of five connection partitions: all three connected, all three separate, or one connected pair and an isolated third terminal. Put three dual terminals on the intervening boundary arcs. Passing to the planar dual graph interchanges the all-connected and all-separate partitions and permutes the three pair partitions. If the network is symmetric under all terminal permutations, equality $P_3=P_0$ of the first two probabilities makes its whole partition law dual-invariant. Independent cells on a self-dual triangular terminal arrangement then give identical primal and dual coarse connection laws. Infinite connectivity is preserved because cells are uniformly finite. For a martini lattice cell with six independent bonds of common parameter $p$, $P_3=3p^5-2p^6$ and $P_3-P_0=(2p^2-1)(p^4-3p^3+2p^2+1)$. The unique zero in $[0,1]$ is $1/\sqrt{2}$.

In bond percolation on the square lattice, the outer boundary of a finite percolation cluster, after filling its holes, consists of closed bonds. Their crossing edges in the planar dual graph form a connected enclosing contour, and hence belong to a single dual bond percolation cluster $K$. If $|K|=m$, both its coordinate widths are at most $m-1$, so the enclosed primal cluster contains at most $m^2$ vertices. A contour enclosing the origin also places every vertex of $K$ within distance $m+1$ of the origin. An exponential volume tail for the subcritical dual bond percolation clusters thus gives a bound $C{e}^{-c\sqrt{n}}$ for a finite primal cluster of size at least $n$.

For a rectangle in a planar lattice, exactly one of an open primal left-to-right crossing and a closed dual top-to-bottom crossing occurs.

For the lattice rectangle with graph vertices $\{0,\dots ,n\}\times \{0,\dots ,n-1\}$, let $H_n$ be an open left-to-right crossing. Its complementary event is a closed dual top-to-bottom crossing. The dual crossing rectangle has width $n-1$ and height $n$, so rotation and translation identify it with the original rectangle. Boundary edges along the starting and ending sides do not affect the crossing event. At $p=1/2$ the primal and dual edge states have identical laws, henceThe one-unit adjustment in the side lengths makes the symmetry exact rather than an informal assertion about a finite graph vertex square.