The clique operator transforms a graph
G into its clique graph
K
(
G
), which is the intersection graph of all the (maximal) cliques of
G. Iterated clique graphs are then defined by
K
n
(
G
)
=
K
(
K
n
−
1
(
G
)
),
K
0
(
G
)
=
G. If there are some
n
≠
m such that
K
n
(
G
)
≅
K
m
(
G
), then we say that
G is clique‐convergent. The clique graph operator and iterated clique graphs have been studied extensively, but no characterization for clique‐convergence has been found so far. Automatic graphs are (not necessarily finite) graphs whose vertices and edges can be recognized by finite automata. Automatic graphs (and automatic structures) have strong decidability properties inherited from the finite automata defining them. Here we prove that clique‐convergence is algorithmically undecidable for the class of automatic graphs. Moreover, the problem remains undecidable, if we reduce to the class that contain only quasi‐clique‐Helly and bounded degree graphs. As a consequence, it follows that clique‐convergence for automatic graphs is not first‐order expressible.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.