Finite FK connectivity-state transfer

Input a finite simple graph $G=(V,E)$, integer $Q\ge 2$, finite activities $v_e\in [0,\mathrm{\infty })$, a total edge order $e_1,\dots ,e_m$, and boundary data consisting of a set $B\subseteq V$ and an equivalence relation $\xi$ on $B$. The relation $\xi$ identifies prescribed boundary vertices before any edge is processed; its singleton classes give free boundary and its non-singleton classes give the wired classes. A vertex incident to no edge is isolated. Put $n_0:=\mathrm{\mid }\{x\in V\setminus B:x\text{ is isolated}\}\mathrm{\mid }$. At time $t$, the active frontier consists of the initially active boundary vertices $B$ together with vertices introduced at a processed edge, minus those already forgotten. A state is a set partition of the active frontier. Its blocks record equivalence generated by the initial boundary relation $\xi$ and processed present edges. A coefficient vector assigns a nonnegative real weight to each such partition.

Initialize the frontier as $B$ with partition $\xi$ and coefficient $Q^{n_0}$; if $B=\mathrm{\varnothing }$, this is the empty partition. For $t=1,\dots ,m$, perform the following operations in order.

1. Introduce each endpoint of $e_t$ that is not yet active as a singleton block.
2. For each state $\pi$ of coefficient $c$, add $c$ to the same state (the edge is absent) and add $c{v}_{e_t}$ to $J_{uv}\pi$ (the edge is present), where $e_t=\{u,v\}$ and $J_{uv}\pi$ merges the blocks containing $u$ and $v$ and leaves all other blocks unchanged. If $u,v$ already share a block, $J_{uv}\pi =\pi$. Thus $J_{uv}^2=J_{uv}$ and the edge operator is $I+v_{e_t}{J}_{uv}$.
3. Forget vertices whose last incident edge is $e_t$. When removing a vertex empties its block, delete that block and multiply its coefficient by $Q$ once; otherwise just remove the vertex from its block. If several vertices in a block are forgotten at this step, the factor $Q$ is applied only when the last active member is removed.

After all edges are processed, forget any remaining active vertices by the same rule and return the coefficient of the empty partition. This includes boundary vertices with no incident edge; each surviving wired component is then counted once. Discrete $\xi$ gives free boundary, while non-singleton classes implement the stated wired boundary condition. A planar-strip annotation consists of a plane embedding, an ordered frontier list at every update, a noncrossing initial relation $\xi$, and the condition that every reachable connectivity partition is noncrossing. For an ordered frontier, “noncrossing” means that no two different blocks contain alternating points $a<b<c<d$ with $a,c$ in one block and $b,d$ in the other. When $u,v$ are adjacent frontier points $i,i+1$, write $J_i$ for $J_{uv}$; these are the noncrossing connectivity states and adjacent-block joins used for planar transfer. The general procedure stores all set partitions when no such annotation is supplied. All arithmetic is exact; each input has finitely many edge updates and finitely many partitions at every frontier.

Finite Potts and Fortuin–Kasteleyn weights

Finite FK boundary wiring