Is there a sequence of undirected graphs $\{C_n\}_{n\in \mathbb N}$, where each $C_n$ has exactly $n$ vertices and the problem Given $n$ and a graph $G$, is $C_n$ an induced subgraph of $G$?is known to be in class $\mathsf{P}$? (For example, when $C_n=K_n$, this is the NP-complete clique problem.)
Is induced subgraph isomorphism easy on an infinite subclass?
cc.complexity theory;graph theory
If I'm not mistaken your question was answered by Chen-Thurley-Weyer-2008 modulo parameterized complexity assumptions. I didn't read the paper carefully yet, but as far as I understood, there is a dichotomy in the sense that if $C$ is finite then the problem is in $P$, but if $C$ has an infinite number of graphs then the induced subgraph isomorphism is $W[1]$ complete (Corollary 4, page 6). Thus it seems that unless the first level $W[1]$ of the $W$ hierarchy collapses to $FPT$, there is no such an infinite class of graphs whose induced subgraph isomorphism is in $P$. There is another interesting result stating that if $P\neq NP$ then there are classes for which the induced isomorphism is neither in $P$ nor $NP$ complete.