{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:04Z","timestamp":1753893844562,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 $\\mathcal{S}$ 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 Bulitko [Graphs with prescribed environments of the vertices. Trudy Mat. Inst. Steklov., 133:78\u201394, 274, 1973].  In contrast, when the neighborhoods are cycle free, all the problems are in the class $\\mathtt{P}$.  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.<\/jats:p>","DOI":"10.37236\/433","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:59:06Z","timestamp":1578697146000},"source":"Crossref","is-referenced-by-count":0,"title":["Building Graphs from Colored Trees"],"prefix":"10.37236","volume":"17","author":[{"given":"Rachel M.","family":"Esselstein","sequence":"first","affiliation":[]},{"given":"Peter","family":"Winkler","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,11,26]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r161\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r161\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T18:21:39Z","timestamp":1579285299000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r161"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,11,26]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/433","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,11,26]]},"article-number":"R161"}}