subvertices

Subvertices is a term used in graph theory to refer to the vertices that belong to a subgraph of a larger graph. If G = (V, E) is a graph and H = (W, F) is a subgraph of G, with W ⊆ V, then every vertex w ∈ W is a subvertex of G (or of H within G). The concept emphasizes inclusion at the vertex level rather than the overall structure.

In a subgraph, the adjacencies between subvertices are inherited from the original graph. Specifically, for w1, w2 ∈ W, the edge {w1, w2} belongs to F if and only if it belongs to E in G. If H is an induced subgraph on W, then F consists of all edges of G with both endpoints in W. Thus, subvertices retain the local neighborhood information as restricted to the chosen vertex set.

Subvertices are used to study local properties and behaviors within the larger graph. They enable analysis of connectivity, degree sequences, local clustering, and minor-related operations by focusing on smaller vertex sets. The notion is particularly helpful when considering how global properties of G relate to properties of its subgraphs.

Example: Let G have vertices {a, b, c, d} with edges forming a cycle a–b–c–d–a. The subgraph H on vertices {a, b, c} has subvertices a, b, c, and edges a–b and b–c. This illustrates how subvertices arise from selecting a vertex subset and restricting the edge set accordingly.

Note that subvertices are simply the vertices of a subgraph; the term is not universally standard and may be described more commonly as “vertices of a subgraph.”