The Electronic Journal of Combinatorics
We will explore the computational complexity of satisfying certain sets of neighborhood conditions in graphs with various properties. More precisely, fix a radius $\rho$ and let $N(G)$ be the set of isomorphism classes of $\rho$-neighborhoods of vertices of $G$ where $G$ is a graph whose vertices are colored (not necessarily properly) by colors from a fixed finite palette. The root of the neighborhood will be the unique vertex at the "center" of the graph. Given a set S of colored graphs with a unique root, when is there a graph G with N (G) = S? Or N (G) ⊂ S? What if G is forced to be infinite, or connected, or both? If the neighborhoods are unrestricted, all these problems are recursively unsolvable; this follows from the work of Bulitko [Graphs with prescribed environmentsof the vertices. Trudy Mat. Inst. Steklov., 133:78–94, 274, 1973]. In contrast, when the neighborhoods are cycle free, all the problems are in the class P. Surprisingly, if G is required to be a regular (and thus infinite) tree, we show the realization problem is NP-complete (for degree 3 and higher); whereas, if G is allowed to be any finite graph, the realization problem is in P.
Esselstein, Rachel M. and Winkler, Peter, "Building Graphs from Colored Trees" (2010). Open Dartmouth: Faculty Open Access Articles. 2322.