  Given two graphs G and H its 1-{\it join} is the graph obtained by taking the
disjoint union of G and H and adding all the edges between a nonempty subset of
vertices of G and a nonempty subset of vertices of H. In general, composition
operations of graphs has played a fundamental role in some structural results
of graph theory and in particular the 1-join composition has played an
important role in decomposition theorems of several class of graphs such as the
claw-free graphs, the bull-free graphs, the perfect graphs, etc.
  A graph G is called {\it $\alpha$-critical} if $\alpha(G\setminus e)>
\alpha(G)$ for all the edges e of G, where $\alpha(G)$, the {\it stability
number} of G, is equal to the maximum cardinality of a stable set of G, and a
set of vertices M of G is {\it stable} if no two vertices in M are adjacent.
The study $\alpha$-critical graphs is important, for instance a complete
description of $\alpha$-critical graphs would yield a good characterization of
the stability number of G.
  In this paper we give necessary and sufficient conditions that G and H must
satisfy in order to its 1-join will be an $\alpha$-critical graph. Therefore we
get a very useful way to construct basic $\alpha$-critical graphs using the
1-join of graphs.
