labelinduced

Labelinduced is a term used in graph theory and data analysis to describe a substructure of a labeled graph defined by a specific vertex label. Given a graph G = (V, E) with a labeling function l: V → L, where L is a set of labels, the label-induced subgraph for a label a ∈ L is the subgraph Ga = (Va, Ea), where Va = {v ∈ V | l(v) = a} and Ea = { (u, v) ∈ E | u ∈ Va and v ∈ Va }. This captures the portion of the graph consisting of nodes bearing the same label, together with the edges between them.

Construction and relation to induced subgraphs: The label-induced subgraph is a specific case of a vertex-induced subgraph, with the vertex set restricted by a labeling condition. The collection of label-induced subgraphs for all labels partitions the vertex set, in the sense that every vertex appears in exactly one label-induced subgraph (assuming a single-label per vertex).

Properties and analysis: For each label a, the subgraph Ga reflects the internal structure of the labeled group. Properties of interest include whether Ga is connected, its density, component counts, and degree distributions restricted to within-label edges. These per-label metrics can reveal homogeneous clusters or label-driven patterns that are not visible in the full graph.

Applications and variants: Labelinduced subgraphs are used in label-aware graph mining, community detection within label groups, and per-label feature extraction for machine learning on graphs. Variants include soft or multi-label settings, where a vertex may contribute to multiple label-induced subgraphs or where edges are weighted by label similarity.

See also: induced subgraph, labeled graph, attribute-aware graph analysis.