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 G and a partial representation $\mathcal{R'}$ giving some pre-drawn chords that represent an induced subgraph of G. The question is whether one can extend $\mathcal{R'}$ to a representation $\mathcal{R}$ of the entire G, i.e., whether one can draw the remaining chords into a partially pre-drawn representation. Our main result is a polynomial-time algorithm for partial representation extension of circle graphs. To show this, we describe the structure of all representation a circle graph based on split decomposition. This can be of an independent interest.

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