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On the other hand, we show that no class excluding a minor can have all commutative idempotent monoids among its endomorphism monoids. As a by\u2010product, we prove that monoids can be represented by graphs of bounded expansion (reproving a result of Ne\u0161et\u0159il and Ossona de Mendez) and \u2010cancellative monoids can be represented by graphs of bounded degree. Finally, we show that not all completely regular monoids can be represented by graphs excluding topological minor (strengthening a result of Babai and Pultr).<\/jats:p>","DOI":"10.1002\/jgt.23262","type":"journal-article","created":{"date-parts":[[2025,6,11]],"date-time":"2025-06-11T05:46:32Z","timestamp":1749620792000},"page":"223-244","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Endomorphism Universality of Sparse Graph Classes"],"prefix":"10.1002","volume":"110","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8151-2184","authenticated-orcid":false,"given":"Kolja","family":"Knauer","sequence":"first","affiliation":[{"name":"Departament de Matem\u00e0tiques i Inform\u00e0tica Universitat de Barcelona Barcelona Spain"},{"name":"LIS, Aix\u2010Marseille Universit\u00e9, CNRS, and Universit\u00e9 de Toulon Marseille France"}]},{"given":"Gil","family":"Puig i Surroca","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Paris\u2010Dauphine, Universit\u00e9 PSL, CNRS, LAMSADE Paris France"}]}],"member":"311","published-online":{"date-parts":[[2025,6,11]]},"reference":[{"volume-title":"Theorie der endlichen und unendlichen Graphen","year":"1936","author":"K\u0151nig D.","key":"e_1_2_10_2_1"},{"key":"e_1_2_10_3_1","first-page":"239","article-title":"Herstellung von Graphen mit vorgegebener abstrakter Gruppe","volume":"6","author":"Frucht R.","year":"1939","journal-title":"Compositio Mathematica"},{"key":"e_1_2_10_4_1","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1949-033-6"},{"key":"e_1_2_10_5_1","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1957-060-7"},{"key":"e_1_2_10_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/322123.322125"},{"key":"e_1_2_10_7_1","unstructured":"P.Klav\u00edkandP.Zeman \u201cAutomorphism Groups of Geometrically Represented Graphs \u201d in32nd International Symposium on Theoretical Aspects of Computer Science (STACS'15) ed.E. 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