  We introduce a concept of a Goldbach factorization graph (GFG) $F_n$, which
can be constructed for each even integer $n$ greater than 2. We prove that, if
$n$ does not satisfy the binary Goldbach conjecture (BGC), then $F_n$ contains
a special source strongly connected component (exceptional autonomous
component, EAC). We analyse existence and properties of EACs using deductive
and computational approaches. In particular, we prove that there exists exactly
one EAC induced by two vertices. Using computer-aided search, we show that for
$n \leq 10^8$ there are 6 EACs, each inside a different GFG, and they are
located at the relative beginning of the checked range, namely, for
$n\in\{128,1718,1862,1928,2200,6142\}$. Using classic graph algorithms, the
constraint programming method, and metaheuristic approaches, we have prepared a
repository of drawings and some selected properties of the found EACs and GFGs
which contain them. The concept of EAC relates to the BGC, but more generally,
it represents interesting self-conjugation of prime numbers under a relation
which combines addition and multiplication.
