The dual complex of ${\mathcal{M}}_{1,n}({\mathbb{P}}^r,d)$ via the geometry of the Vakil--Zinger moduli space

We study normal crossings compactifications of the moduli space of maps ${\mathcal{M}}_{g,n}({\mathbb{P}}^r,d)$, for $g=0$ and $g=1$. In each case we explicitly determine the dual boundary complex, and prove that it admits a natural interpretation as a moduli space of decorated metric graphs. We prove that the dual complexes are contractible when $r\ge 1$ and $d>g$. When $g=1$, our result depends on a new understanding of the connected components of boundary strata in the Vakil–Zinger desingularization and its modular interpretation by Ranganathan–Santos-Parker–Wise.