TITLE: Difference between a sub graph and induced sub graph.
QUESTION [40 upvotes]: I have the following paragraph in my notes:

If $G=(V,E)$ is a general graph . Let $U\subseteq V$ and let $F$ be a subset of $E$ such that the vertices of each edge in $F$ are in $U$ ,
  then $H=(U,F)$ is also a general graph and $H$ is a subgraph of $G$ .  
If $F$ consists of all edges of $G$ which have endpoints in $U$ ,then $H$ is called induced subgraph of $G$ and is denoted by $G_U. $  

From here both the definition of a subgraph and a induced subgraph seem same to me..
I can't understand what is the difference between them...
Please help with this..

REPLY [2 votes]: Say you have a graph $G(V,E,\phi)$ now any graph $G'(V',E',\phi')$ can be a sub-graph of G only if it satisfies all the 3 conditions

$V' \subseteq V$
$E' \subseteq E$
$\phi'(e)=\phi(e)$ for all e belonging E'
While for constructing an induces subdigraph you first take a subset say W of vertex set V, then draw all the edges that lie between all those vertices(unlike a subgraph here you don't have the option to not chose an edge) be careful not to choose an edge that has one vertex in W but one not belonging to W.