  Ara\'ujo, Kinyon and Konieczny (2011) pose several problems concerning the
construction of arbitrary commuting graphs of semigroups.
  We observe that every star-free graph is the commuting graph of some
semigroup. Consequently, we suggest modifications for some of the original
problems, generalized to the context of families of semigroups with a bounded
number of generators, and pose related problems.
  We construct monomial semigroups with a bounded number of generators, whose
commuting graphs have an arbitrary clique number. In contrast to that, we show
that the diameter of the commuting graphs of semigroups in a wider class
(containing the class of nilpotent semigroups), is bounded by the minimal
number of generators plus two.
  We also address a problem concerning knit degree.
