  A graph that can be generated from $K_1$ using joins and 0-sums is called a
cograph. We define a sesquicograph to be a graph that can be generated from
$K_1$ using joins, 0-sums, and 1-sums. We show that, like cographs,
sesquicographs are closed under induced minors. Cographs are precisely the
graphs that do not have the 4-vertex path as an induced subgraph. We obtain an
analogue of this result for sesquicographs, that is, we find those
non-sesquicographs for which every proper induced subgraph is a sesquicograph.
