  The complexity of graph isomorphism (GraphIso) is a famous unresolved problem
in theoretical computer science. For graphs $G$ and $H$, it asks whether they
are the same up to a relabeling of vertices. In 1981, Lubiw proved that list
restricted graph isomorphism (ListIso) is NP-complete: for each $u \in V(G)$,
we are given a list ${\mathfrak L}(u) \subseteq V(H)$ of possible images of
$u$. After 35 years, we revive the study of this problem and consider which
results for GraphIso translate to ListIso.
  We prove the following: 1) When GraphIso is GI-complete for a class of
graphs, it translates into NP-completeness of ListIso. 2) Combinatorial
algorithms for GraphIso translate into algorithms for ListIso: for trees,
planar graphs, interval graphs, circle graphs, permutation graphs, bounded
genus graphs, and bounded treewidth graphs. 3) Algorithms based on group theory
do not translate: ListIso remains NP-complete for cubic colored graphs with
sizes of color classes bounded by 8.
  Also, ListIso allows to classify results for the graph isomorphism problem.
Some algorithms are robust and translate to ListIso. A fundamental problem is
to construct a combinatorial polynomial-time algorithm for cubic graph
isomorphism, avoiding group theory. By the 3rd result, ListIso is NP-hard for
them, so no robust algorithm for cubic graph isomorphism exists, unless P = NP.
