Building graphs from colored trees
Summary: We 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 is the unique vertex at the ``center'' of the graph. Given a set \({\mathcal S}\) of colored graphs with a unique root, when is there a graph \(G\) with \(N(G)={\mathcal S}\)? Or \(N(G)\subset{\mathcal S}\)? 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 \textit{V. K. Bulitko} [Tr. Mat. Inst. Steklova 133, 78--94 (1973; Zbl 0295.05124)]. 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.
scientific article; zbMATH DE number 5827410