Geodesic
Congruences and Hoehnke Radicals on Graphs
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 4, pp. 1067-1084

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We motivate, introduce, and study radicals on classes of graphs. This concept, and the theory which is developed, imitates the original notion of a Hoehnke radical in universal algebra using congruences. It is shown how this approach ties in with the existing theory of connectednesses and disconnectednesses (= Kurosh-Amitsur radical theory).
Keywords: congruences and quotients of graphs, Hoehnke radicals of graphs, connectednesses and disconnectednesses of graphs, Kurosh-Amitsur radicals of graphs, subdirect representations of graphs 
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Broere, Izak; Heidema, Johannes; Veldsman, Stefan. Congruences and Hoehnke Radicals on Graphs. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 4, pp. 1067-1084. http://geodesic.mathdoc.fr/item/DMGT_2020_40_4_a8/