  We further develop the theory of $W\!$-graph ideals, first introduced by the
authors in reference [6]. We discuss $W\!$-graph subideals, and induction and
restriction of $W\!$-graph ideals for parabolic subgroups. We introduce
$W\!$-graph biideals: those $W\!$-graph ideals that yield $(W\times W^{\mathrm
o})$-graphs, where $W^{\mathrm o}$ is the group opposite to $W$. We determine
all $W\!$-graph ideals and biideals in finite Coxeter groups of rank 2.
