What is an induced subgraph?

An induced subgraph is a graph that is obtained by removing some vertices and edges from a larger graph, while preserving the connections between the remaining vertices. Specifically, an induced subgraph of a graph G is a new graph H that comprises a subset of the vertices of G (called the vertex subset) and all the edges of G that have both endpoints in this subset. In other words, a subgraph H of G is induced if and only if for every two vertices u and v in H, there is an edge between them in H if and only if there is an edge between them in G.

An induced subgraph is often used to study smaller and simpler graphs that retain some key properties of the larger graph. It can help in reducing the complexity of a graph by considering only a relevant subset of its vertices and edges. For instance, an induced subgraph can be used to identify cliques (fully-connected subgraphs) or other important substructures in a graph. It can also be used to study the relationship between the structural properties of a graph and its function or behavior, such as in network analysis or graph algorithms.