Avertices

Avertices is a term used in graph theory to describe the set of vertices that are related to a distinguished subset A of the vertex set V in a graph G = (V, E). The most common formalization is the A-neighborhood Av(A), defined as Av(A) = { v ∈ V \ A : ∃ a ∈ A with {v, a} ∈ E }. In some conventions, vertices in A are also considered A-vertices, leading to the closed A-neighborhood N[A] = A ∪ Av(A). The concept is used to study how the subset A interacts with the rest of the graph, including boundary size and expansion properties.

Variants and properties of Avertices include the open A-vertex set Av(A), the closed A-vertex set N[A], and the A-vertex boundary ∂A = Av(A). These notions underpin several measures such as A-vertex expansion, which gauges how many A-vertices are adjacent to a given subset relative to its size. Relationships exist between Avertices and domination, as A-dominated subsets influence the reach of Av(A) across the graph. Monotonicity holds in the sense that enlarging A typically enlarges Av(A) or N[A], and under certain conditions Av(A) can be used to construct subgraphs or partition schemes.

Applications of Avertices appear in network analysis and algorithm design, including community detection, influence propagation studies, and robustness analysis. They also appear in spectral graph theory where A-vertex sets help define specialized operators or boundary conditions for analysis on graphs.

Example: in a path graph with vertices v1–v5 and A = {v2}, Av(A) = {v1, v3}, and N[A] = {v1, v2, v3}. This illustrates how Avertices capture the immediate surroundings of a distinguished subset.

See also: neighborhoods, closed neighborhood, boundary in graphs, vertex expansion, domination. The concept is widely used in theoretical investigations and practical network problems.