Induced subgraph

In the

Formally, let ${\displaystyle G=(V,E)}$ be any graph, and let ${\displaystyle S\subseteq V}$ be any subset of vertices of G. Then the induced subgraph ${\displaystyle G[S]}$ is the graph whose vertex set is ${\displaystyle S}$ and whose edge set consists of all of the edges in ${\displaystyle E}$ that have both endpoints in ${\displaystyle S}$.[1] That is, for any two vertices ${\displaystyle u,v\in S}$, ${\displaystyle u}$ and ${\displaystyle v}$ are adjacent in ${\displaystyle G[S]}$ if and only if they are adjacent in ${\displaystyle G}$. The same definition works for

• shortest path between any two vertices in an unweighted graph is always an induced path, because any additional edges between pairs of vertices that could cause it to be not induced would also cause it to be not shortest. Conversely, in distance-hereditary graphs, every induced path is a shortest path.^[2]
• Induced cycles are induced subgraphs that are cycles. The girth of a graph is defined by the length of its shortest cycle, which is always an induced cycle. According to the strong perfect graph theorem, induced cycles and their complements play a critical role in the characterization of perfect graphs.^[3]
• edgeless graphs
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• Induced matchings are induced subgraphs that are matchings.
• The neighborhood of a vertex is the induced subgraph of all vertices adjacent to it.

Computation

The