AbstractThere is a well-developed theory of connectednesses and disconnectednesses (= radical theory) for the category of graphs that admit loops. Here it is shown that such a theory for the category of graphs that do not allow loops degenerates to the trivial case for all the Hoehnke radicals, but there are non-trivial connectednesses (KA-radical classes) and disconnectednesses (KA-semisimple classes). Moreover, the connectednesses and disconnectednesses always come as complementary pairs.