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We study graphs of (generalized) joins and intersections of finitely generated subgroups of a free group. We show how to disprove a lemma of Imrich and Müller on these graphs, how to repair the lemma and how to utilize it.
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Sergei V. Ivanov, On joins and intersections of subgroups in free groups. J. Comb. Algebra 2 (2018), no. 1 pp. 1–18