Hardness of Subgraph isomorphism problem for sparse pattern graph
It is $W[1]$-hard even when $G$ has maximum degree $3$, but $FPT$ if $G$ has constant treewidth (all the above examples have constant treewidth). See the paper Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask) by Marx and Pilipczuk. 

Indeed, it follows from an earlier paper of Marx: Can You Beat Treewidth? Theory of Computing 6(1): 85-112 (2010) that a $f(k)n^{o(k/\log k)}$
time algorithm would violate the ETH. Here $n$ is the number of vertices in $H$ and $k$ is the number of vertices in $G$. This holds even when $G$ has max degree $3$ (but the degree of $H$ is unbounded).