let H be minor of a graph G. Is it always true that embedding surface of H is less than embedding surface of G

Question

let $H$ be minor of a graph $G$. Is it always true that (genus of embedding surface of $H$) is $\leq$ (genus of embedding surface of $G$).

Answer 1

Consider the embedding of $G$. I claim we can construct an embedding of $H$ on the same surface. For the edge deletions this is trivial. For the edge contractions one can picture the process as a continuous deformation of the graph without changing the surface. Hence any sequence of edge deletions and and edge contractions that builds $H$ from $G$ gives you an embedding of $H$ on the embedding surface of $G$. Therefore the embedding genus of $H$ has to be smaller or equal than the embedding genus of $G$.