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.keywordsintersection graphinterval graphchordal graphprime graphcircle graphthese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.