Suppose you have a group $G$ given by a finite presentation $\langle X; \mathbf{r}\rangle$, and also suppose you know that this group decomposes as a free product with amalgamation $A\ast_CB$ (with $A$, $B$ and $C$ all infinite, and I am happy to assume $C\cong\mathbb{Z}$ for simplicity).
Is there any way we can link the presentation complex $\mathcal{C_P}$ of $G$ (single vertex, edges labelled from $X$, 2-cells attached using $\mathbf{r}$) with the Seifert-van Kampen-esque 2-complex $\mathcal{C}_{SvK}$ underlying $A\ast_CB$?
My initial thought was to look at the cover associated to the amalgamating subgroup $C$, and ask if I get a complex with a subcomplex which satisfies Seifert-van Kampen. However, this doesn't seem to work even if I take the cover of the presentation complex of $A\ast_CB$.