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The energy of a configuration is given by the weighted sum of the areas of the interfaces between the different regions and a general volume-order term. We establish existence of minimizing configurations via relaxation of the energy involved, in any dimension. Moreover, we prove partial regularity results for volume-constrained minimizers in two space dimensions. Thin films of diblock copolymers are a possible application and motivation for considering this problem.<\/jats:p>","DOI":"10.1007\/s00332-022-09852-3","type":"journal-article","created":{"date-parts":[[2022,10,5]],"date-time":"2022-10-05T18:19:27Z","timestamp":1664993967000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Area Quasi-minimizing Partitions with a Graphical Constraint: Relaxation and Two-Dimensional Partial Regularity"],"prefix":"10.1007","volume":"32","author":[{"given":"Marco","family":"Bonacini","sequence":"first","affiliation":[]},{"given":"Riccardo","family":"Cristoferi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,10,5]]},"reference":[{"key":"9852_CR1","doi-asserted-by":"publisher","first-page":"515","DOI":"10.1007\/s00220-013-1733-y","volume":"322","author":"E Acerbi","year":"2013","unstructured":"Acerbi, E., Fusco, N., Morini, M.: Minimality via second variation for a nonlocal isoperimetric problem. 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