We investigate some graph classes that lie on complexity dividing lines in computer science and show using that the lines mostly hold when graphs are countably infinite. If graphs are finite, we may consider a class complex if it is graph-isomorphism complete, which means that maximal time is needed to check if two graphs in the class are isomorphic. If graphs are allowed to be countable, we may consider a class complex if it is universal under effective bi-interpretabilty, which means that one can effectively code any given countable graph $G$ as a graph $H$ in the class and also effectively recover $G$ from $H$. When restricting our study to $\mathrm{free}(F)$, which is the class of graphs that cannot embed a fixed finite graph $F$ as an induced subgraph, we show that $\mathrm{free}(F)$ is graph-isomorphism complete if and only if it is universal. The statement holds even if universality is replaced by not being $\Sigma$-small, or by not having the computable embeddability condition, which are other ways to assess the complexity of classes of countable graphs. But the statement fails when universality is replaced by Borel-completeness, where the coding and recovery of $G$ only need to be performed by Borel functions, not necessarily computable ones. There are many future directions, including investigating whether GI-completeness always aligns with universality for other hereditary graph classes.