outdegvC0

OutdegvC0 is a notational construct used in graph theory to express the out-degree of a vertex relative to a specified subset of vertices, commonly denoted as C0. It appears in discussions of directed graphs where directional reachability into a particular subcollection of vertices is of interest.

In a directed graph G = (V, E), for a vertex v ∈ V and a subset C0 ⊆ V, the quantity outdegvC0 is defined as deg+(v; C0) = |{u ∈ C0 : (v, u) ∈ E}|, the number of edges leaving v and terminating in C0. Variants may write outdegC0(v) or outdeg(v, C0). When C0 = V, outdegvC0 reduces to the standard out-degree deg+(v).

Typical uses include analyzing how a vertex connects to a particular subcollection of vertices, such as evaluating boundary reach in community detection, influence spread models, or subgraph sampling. The measure helps compare how different vertices contribute to crossing into C0.

Computationally, for each vertex v, one counts the outgoing neighbors that lie in C0. If the graph is stored with adjacency lists and C0 membership is marked (for example, with a boolean array), the count can be obtained in time proportional to the number of outgoing edges from v. Computing the value for all vertices costs O(m), where m is the number of edges, plus the overhead of maintaining the C0 indicator.

Example: consider a directed graph with vertices {a, b, c, d} and edges {a→b, a→c, b→c, c→d, d→a}. Let C0 = {b, c}. Then outdegvC0(a) = 2 (to b and c), outdegvC0(b) = 1 (to c), outdegvC0(c) = 0, and outdegvC0(d) = 0. See also deg+, adjacency, induced subgraph.