Extending partial representations of circle graphs

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 and a partial representation giving some predrawn chords that represent an induced subgraph of . The question is whether one can extend to a representation of the entire graph , that is, whether one can draw the remaining chords into a partially predrawn representation to obtain a representation of . Our main result is an time algorithm for partial representation extension of circle graphs, where 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.