  Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it
contains no subgraph isomorphic to $H_1$ or $H_2$. We continue a recent study
into the clique-width of $(H_1,H_2)$-free graphs and present three new classes
of $(H_1,H_2)$-free graphs of bounded clique-width and one of unbounded
clique-width. The four new graph classes have in common that one of their two
forbidden induced subgraphs is the diamond (the graph obtained from a clique on
four vertices by deleting one edge). To prove boundedness of clique-width for
the first three cases we develop a technique based on bounding clique covering
number in combination with reduction to subclasses of perfect graphs. We extend
our proof of unboundedness for the fourth case to show that Graph Isomorphism
is Graph Isomorphism-complete on the same graph class. We also show the
implications of our results for the computational complexity of the Colouring
problem restricted to $(H_1,H_2)$-free graphs.
