Knauer, KoljaPuig i Surroca, G.2026-02-202026-02-202025-07-010364-9024https://hdl.handle.net/2445/227127We show that every commutative idempotent monoid (a.k.a. lattice) is the endomorphism monoid of a subcubic graph. This solves a problem of Babai and Pultr and the degree bound is best-possible. 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-product, we prove that monoids can be represented by graphs of bounded expansion (reproving a result of Nešetřil and Ossona de Mendez) and $k$-cancellative 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).22 p.application/pdfengcc by-nc-nd (c) Kolja Knauer et al., 2025http://creativecommons.org/licenses/by-nc-nd/4.0/Isomorfismes (Matemàtica)Teoria de grafsRepresentacions de semigrupsIsomorphisms (Mathematics)Graph theoryRepresentations of semigroupsinfo:eu-repo/semantics/article7659602026-02-20info:eu-repo/semantics/openAccess