It is well known that the subgraph isomorphism problem is NP-complete. And so a polynomial-time algorithm for solving it would mean P = NP. Thus I'm interested in whether a bounded version of the problem has a polynomial time algorithm that is known about. Many applications of graphs don't require more than say 50 incident edges per node. For example in visualizations involving commutative diagrams of mathematics, rarely is there a need for anywhere close to that number of connections. This is because as the graph complexity grows, the diagram becomes more useless as a mnemonic for studying math.
So, my question is: if we constrain our nodes to have maximally a constant number of incident edges, then does the subgraph isomorphism problem become polynomial-time solvable?
That is, both the query graph $G$ and the large graph $H$ have bounded incident edges, such that you're trying to find an isomorphism of $G \xrightarrow{\sim} H' \subset H$.