Definition: Subgraphs and Supergraphs; Induced Subgraph

Let \(G(V,E,\gamma)\) be an undirected graph. A graph \(S(V',E',\gamma')\) is called a subgraph of \(G\), written as \(S\subseteq G\), if it fulfills the following properties:

  1. \(V'\) consists only of vertices from \(V\), formally \(V'\subseteq V\).
  2. \(E'\) consists only of edges from \(E\), formally \(E'\subseteq E\).
  3. \(\gamma'\) is the restriction of \(\gamma\) on \(E'\), i.e. \(\gamma':={\gamma|}_{E'}E\mapsto V\). (Note: This property is not necessary for a simple undirected graph \(G(V,E)\)).

If \(S\) is a subgraph of \(G\), then \(G\) is called the supergraph of \(S\).

If \(S\) contains all the edges \(e\) with \(\gamma(e)\in V'\), then \(S\) is an induced subgraph of \(G\); we say that the vertices \(V'\) induce or span \(S\) in \(G\) and write \(S:=G[V']\). Thus, if \(V'\subseteq V\) is any set of vertices, \(V'=\{v_1,v_2,\ldots,v_n\}\), then the induced subgraph \(G[V']=G[v_1,v_2,\ldots,v_n]\) denotes the graph whose edges are precisely the edges of \(G\) with ends in \(V'\).



In the above figure, the graphs \(G_2, G_3\) and \(G_4\) are all subgraphs of the graph \(G_1\). However, only \(G_2\) and \(G_3\) are induced subgraphs of \(G_1\), since

\[G_2=G_1[a,c,d],\quad G_3=G_1[a,b,c],\]


\[G_4\neq G_1[a,b,c,d].\]

  1. Definition: Spanning Subgraph

Algorithms: 1
Definitions: 2 3 4 5 6
Lemmas: 7
Proofs: 8 9 10 11 12 13
Propositions: 14 15

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  1. Diestel, Reinhard: "Graph Theory, 3rd Edition", Springer, 2005