The first subquestion is "has a standard notion of semidirect product been defined in graph theory"? If yes, i'd like to know if the definition i'm gonna give is equivalent to the standard one. I'd also like to know if there's some litterature about it. If the answer was "no", would you accept my definition as a reasonable "semidirect product"?
Consider two graphs $G$ and $H$ (finite, undirected, simple, loopless, and of order $n,m$ respectively), and let $Aut(H)$ denote the set of automorphisms of $H$. Pick $\phi_1, ..., \phi_n \in Aut(H)$, and construct a semidirect product as follows:
Label the elements of $G$ as $1,2,...,n$. Pick $n$ copies $H_1, ..., H_n$ of $H$. If $x_i \in H_i$ and $x_j \in H_j$, add an edge between them iff $(i,j) \in E_G$ and $\phi_i(x_i)= \phi_j(x_j)$. The result of this operation is the semidirect product $G \ltimes_{(\phi_1, ..., \phi_n)} H$
I hope this is clear... In other words, you perform a simple cartesian product, but you tilt each "$H$" component using an automorphism before connecting the respectives fibers. As an exemple, consider the Petersen graph as a semidirect product of $K_2$ and $C_5$ (where we can choose the identity and a nontrivial automorphism):
Let $G = K_2$ and let $H=H_1=H_2$ be the cycle $a - b - c - d - a$ (dashes mean adiacency. Also let $\phi_1, \phi_n$ be automorphisms in $Aut(H)$ defined as:
$\phi_1 = Id$
$\phi_2(a)=a; \phi_2(b)=c; \phi_2(c)=e; \phi_2(d)=b; \phi_2(e)=d;$
Following the definition we will add an edge between:
$a \in H_1$ and $a \in H_2$ as $a = \phi_1(a) = \phi_2(a) = a$
$b \in H_1$ and $d \in H_2$ as $b = \phi_1(b) = \phi_2(d) = b$
$c \in H_1$ and $b \in H_2$ as $c = \phi_1(c) = \phi_2(b) = c$
$d \in H_1$ and $e \in H_2$ as $c = \phi_1(d) = \phi_2(e) = d$
$e \in H_1$ and $c \in H_2$ as $c = \phi_1(e) = \phi_2(c) = e$
Thanks for reading. Feedback would be really appreciated.