Discrete Dirichlet problem

Remarks

The problem always has exactly one solution. Uniqueness follows from the maximum principle: the difference $h$ of two solutions is harmonic on $I$ and zero on $B$. At an interior vertex where $h$ is largest, harmonicity forces the same value at all neighbours. Connectedness spreads this value to a vertex of $B$, so $\max h = 0$, and in the same way $\min h = 0$. The conditions form a square system of $\abs{I}$ linear equations in the $\abs{I}$ unknowns $f(x)$, $x \in I$, so uniqueness gives existence. The solution has the probabilistic form $f(x) = \mathbb{E}_x\big[g(X_\tau)\big]$, where $(X_n)$ is the random walk with $p(x, y) = c(x, y)/c(x)$ started at $x$ and $\tau$ is its first hitting time of $B$. Example: on the path with vertices $0, 1, \dots, n$, unit weights and $B = \set{0, n}$, the solution is the linear interpolation $f(k) = g(0) + \tfrac{k}{n}\big(g(n) - g(0)\big)$. Connectedness matters: if a component of the graph does not meet $B$, any constant can be added to $f$ on that component, and the solution is not unique.