The subgraph isomorphism problem problem is to determine given $G$ and $H$ whether $G$ is a subgraph of $H$.
Let $G$ and $H$ be regular graphs with degree of $H$ greater than degree of $G$.
Does the subgraph isomorphism problem remain NP-complete for the following case:
$1.$ Girth of $G$ and $H$ are fixed, say $girth=3$ or a fixed $h$?
How large should a fixed $k$-diameter $d_H$-regular graph $H$ be for it to have a fixed $k$-diameter $d_G$-regular subgraph $G$ of vertex count $n_G$ when both have the same girth?