Abstract
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.~\cite{JEN12}, Lockett and Truss \cite{LOC14} as well as Rose~\cite{ROS11} 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.
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