Classification of countable 2-colored ultrahomogeneous graphs where each color class forms a disjoint union of cliques

We classify the countable ultrahomogeneous 2-vertex-colored graphs in which the color classes form disjoint unions of cliques. This generalizes work by Jenkinson et. al. (2012), Lockett and Truss (2014) as well as Rose (2011) on ultrahomogeneous $n$-graphs. As the key aspect in such a classification, we identify a concept called piecewise ultrahomogeneity. We prove that there are two specific graphs whose occurrence essentially dictates whether a graph is piecewise ultrahomogeneous, and we exploit this fact to prove the classification.