# NP hardness through Restriction

Let's say I have a decision problem $P$ on graphs for which I know that it is NP-hard on graphs with maximum degree $d$. Does this then imply that it is NP-hard on $d$-regular graphs? Although it might seem obviously true, maybe it is inherent in the reduction to show that $P$ is hard, that some vertices have degree less than $d$.

In your case, the $d$-regular graphs are a special case of graphs of max degree $d$, so NP-Hardness for max degree $d$ is not an automatic proof of NP-Hardness of $d$ regular graphs.
Here's a reasonably natural one: on an input $(G,k)$, determine whether $G$ has a connected regular subgraph with at least $k$ edges. For 3-regular graphs this is trivial, but if max degree is 3 and the input is connected, not a tree, and not regular, then the largest such subgraph is the longest cycle, so the problem is NP-complete.