  Intersection graphs are well-studied in the area of graph algorithms. Some
intersection graph classes are known to have algorithms enumerating all
unlabeled graphs by reverse search. Since these algorithms output graphs one by
one and the numbers of graphs in these classes are vast, they work only for a
small number of vertices. Binary decision diagrams (BDDs) are compact data
structures for various types of data and useful for solving optimization and
enumeration problems. This study proposes enumeration algorithms for five
intersection graph classes, which admit $\mathrm{O}(n)$-bit string
representations for their member graphs. Our algorithm for each class
enumerates all unlabeled graphs with $n$ vertices over BDDs representing the
binary strings in time polynomial in $n$. Moreover, our algorithms are extended
to enumerate those with constraints on the maximum (bi)clique size and/or the
number of edges.
