Boundary of a finite region

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

Boundary of a finite region

Let $\Lambda$ be a finite subset of vertices in a graph. In the lattice setting, take $\Lambda\subset \mathbb{Z}^d$ with adjacency given by nearest-neighbor-zd .

Write $x\sim y$ if $x$ and $y$ are adjacent.

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

\partial \Lambda := \{y\notin \Lambda : \exists x\in \Lambda \text{ such that } x\sim y\}.

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

Inner (internal) vertex boundary. The inner boundary is

\partial^{-}\Lambda := \{x\in \Lambda : \exists y\notin \Lambda \text{ such that } x\sim y\}.

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

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

\delta \Lambda := \{\{x,y\} : x\in \Lambda,\ y\notin \Lambda,\ x\sim y\}.

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.