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We classify the countable ultrahomogeneous 2-vertex-colored graphs in whichthe color classes are imprimitive, i.e., up to complementation they formdisjoint unions of cliques. This generalizes work by Jenkinson, Lockett andTruss as well as Rose on ultrahomogeneous $n$-graphs. As the key aspect in sucha classification, we identify a concept called piecewise ultrahomogeneity. Weprove that there are two specific graphs whose occurrence essentially dictateswhether a graph is piecewise ultrahomogeneous, and we exploit this fact toprove the classification.

Countable ultrahomogeneous graphs on two imprimitive color classes