  The Lachlan-Woodrow Theorem identifies ultrahomogeneous graphs up to
isomorphism. Recently, the present author and D. Hartman classified
MB-homogeneous graphs up to bimorphism-equivalence. We extend those results in
this paper, showing that every IB-homogeneous graph is either ultrahomogeneous
or MB-homogeneous, and thus all the IB-homogeneous graphs are known up to
bimorphism-equivalence.
