Alternating maps on Hatcher-Thurston graphs

Let $S_{1}$ and $S_{2}$ be connected orientable surfaces of genus $g_{1}, g_{2} \geq 3$, $n_{1},n_{2} \geq 0$ punctures, and empty boundary. Let also $\varphi: \mathcal{HT}(S_{1}) \rightarrow \mathcal{HT}(S_{2})$ be an edge-preserving alternating map between their Hatcher-Thurston graphs. We prove that $g_{1} \leq g_{2}$ and that there is also a multicurve of cardinality $g_{2} - g_{1}$ contained in every element of the image. We also prove that if $n_{1} = 0$ and $g_{1} = g_{2}$, then the map $\widetilde{\varphi}$ obtained by filling the punctures of $S_{2}$, is induced by a homeomorphism of $S_{1}$.