  This paper analyses the construction of the kernel graph of a
non-synchronizing transformation semigroup and introduces the inverse
synchronization problem. Given a transformation semigroup $S\leq T_n$, we
construct the kernel graph $\text{Gr}(S)$ by saying $v$ and $w$ are adjacent,
if there is no $f\in S$ with $vf=wf$. The kernel graph is trivial or complete
if the semigroup is a synchronizing semigroup or a permutation group,
respectively. The connection between graphs and synchronizing (semi-) groups
was established by Cameron and Kazanidis, and it has led to many results
regarding the classification of synchronizing permutation groups, and the
description of singular endomorphims of graphs. This paper, firstly, emphasises
the importance of this construction mainly by proving its superior structure,
secondly, analyses the construction and discusses minimal generating sets and
their combinatorial properties, and thirdly, introduces the inverse
synchronization problem. The third part also includes an additional
characterization of primitive groups.
