# Induced subgraph

In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges connecting pairs of vertices in that subset.

## Definition

Formally, let G = (V, E) be any graph, and let SV be any subset of vertices of G. Then the induced subgraph G[S] is the graph whose vertex set is S and whose edge set consists of all of the edges in E that have both endpoints in S. The same definition works for undirected graphs, directed graphs, and even multigraphs.

The induced subgraph G[S] may also be called the subgraph induced in G by S, or (if context makes the choice of G unambiguous) the induced subgraph of S.

## Examples

Important types of induced subgraphs include the following.

## Computation

The induced subgraph isomorphism problem is a form of the subgraph isomorphism problem in which the goal is to test whether one graph can be found as an induced subgraph of another. Because it includes the clique problem as a special case, it is NP-complete.