  A self-contained graph is an infinite graph which is isomorphic to one of its
proper induced subgraphs. In this paper, these graphs are studied by presenting
some examples and defining some of their sub-structures such as removable
subgraphs and the foundation. Then, we show that the general version of graph
alternative conjecture, which says every graph has infinitely many strong twins
or none, can be deduced from its connected version, which says every connected
graph has infinitely many connected strong twins or none. Moreover, we try to
find out under what conditions on two arbitrary removable subgraphs, their
union is also a removable subgraph.
