Knowlpedia

Boundary of a finite region

Standard notions of boundary for a finite subset of a graph or lattice.

Let Λ\Lambda be a finite subset of vertices in a graph. In the lattice setting, take Λ⊂Zd\Lambda\subset \mathbb{Z}^d with adjacency given by .

Write x∼yx\sim y if xx and yy are adjacent.

Outer (external) vertex boundary. The outer boundary of Λ\Lambda is

∂Λ:={y∉Λ:∃x∈Λ such that x∼y}.\partial \Lambda := \{y\notin \Lambda : \exists x\in \Lambda \text{ such that } x\sim y\}.

These are the vertices outside Λ\Lambda that are one step away from Λ\Lambda.

Inner (internal) vertex boundary. The inner boundary is

∂−Λ:={x∈Λ:∃y∉Λ such that x∼y}.\partial^{-}\Lambda := \{x\in \Lambda : \exists y\notin \Lambda \text{ such that } x\sim y\}.

These are the vertices inside Λ\Lambda that have at least one neighbor outside.

Edge boundary. The edge boundary (also called the set of cut edges) is

δΛ:={{x,y}:x∈Λ, y∉Λ, x∼y}.\delta \Lambda := \{\{x,y\} : x\in \Lambda,\ y\notin \Lambda,\ x\sim y\}.
Remarks

Different authors may use different boundary conventions (vertex vs. edge boundary, inner vs. outer), so it is good practice to check which version is intended in a given argument.