  Circle graphs are intersection graphs of chords of a circle. In this paper,
we present a new algorithm for the circle graph isomorphism problem running in
time $O((n+m)\alpha(n+m))$ where $n$ is the number of vertices, $m$ is the
number of edges and $\alpha$ is the inverse Ackermann function. Our algorithm
is based on the minimal split decomposition [Cunnigham, 1982] and uses the
state-of-art circle graph recognition algorithm [Gioan, Paul, Tedder, Corneil,
2014] in the same running time. It improves the running time $O(nm)$ of the
previous algorithm [Hsu, 1995] based on a similar approach.
