  The partial representation extension problem is a recently introduced
generalization of the recognition problem. A circle graph is an intersection
graph of chords of a circle. We study the partial representation extension
problem for circle graphs, where the input consists of a graph $G$ and a
partial representation $\cal R'$ giving some pre-drawn chords that represent an
induced subgraph of $G$. The question is whether one can extend $\cal R'$ to a
representation $\cal R$ of the entire graph $G$, i.e., whether one can draw the
remaining chords into a partially pre-drawn representation to obtain a
representation of $G$. Our main result is an $O(n^3)$ time algorithm for
partial representation extension of circle graphs, where $n$ is the number of
vertices. To show this, we describe the structure of all representations of a
circle graph using split decomposition. This can be of independent interest.
