Induced subgraph

An induced subgraph is formed by choosing vertices from a graph and retaining every original edge whose endpoints are both chosen.

Graph (discrete mathematics): A graph is a mathematical structure consisting of vertices joined by edges. An induced subgraph inherits its vertices and edges from this broader structure.
Induced subgraph on a vertex set: The induced subgraph on a specified vertex set contains those vertices and every original edge between them. This names the construction once its selecting vertex set is specified.
Edge-induced subgraph: An edge-induced subgraph is formed from a chosen edge set together with its incident vertices. It starts with edges, whereas an induced subgraph starts with vertices.
Neighborhood (graph theory): The neighborhood of a vertex is the set of vertices adjacent to it. The subgraph induced by a neighborhood captures the vertex's local connections.
Vertex-induced subgraph: A subgraph obtained by selecting a vertex set and retaining all edges between its selected vertices. This is the standard construction described by the concept.
Adjacency matrix: An adjacency matrix records which pairs of vertices in a graph are connected by edges. Restricting it to selected vertices gives the induced subgraph's adjacency matrix.
Spanning subgraph: A spanning subgraph contains every vertex of a graph but may omit edges. It fixes the vertex set and permits edge deletion, unlike vertex induction.
Girth (graph theory): The girth of a graph is the length of its shortest cycle. Cycles appear as induced subgraphs when they have no extra edges among their vertices.
Vertex subset: A subset is a collection of elements selected from a larger set. The chosen vertices determine the induced subgraph's entire vertex set.
Graph complement: The complement of a simple graph has the same vertices, with edges exactly where the original has none. Taking an induced subgraph commutes with complementation on the same selected vertices.

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