  A biclique of a graph $G$ is an induced complete bipartite subgraph of $G$
such that neither part is empty. A star is a biclique of $G$ such that one part
has exactly one vertex. The star graph of $G$ is the intersection graph of the
maximal stars of $G$. A graph $H$ is star-critical if its star graph is
different from the star graph of any of its proper induced subgraphs. We begin
by presenting a bound on the size of star-critical pre-images by a quadratic
function on the number of vertices of the star graph, then proceed to describe
a Krausz-type characterization for this graph class; we combine these results
to show membership of the recognition problem in \textsf{NP}. We also present
some properties of star graphs. In particular, we show that they are
biconnected, that every edge belongs to at least one triangle, characterize the
structures the pre-image must have in order to generate degree two vertices,
and bound the diameter of the star graph with respect to the diameter of its
pre-image. Finally, we prove a monotonicity theorem, which we apply to list
every star graph on at most eight vertices.
