Extending Partial Representations of Circle Graphs in Near-Linear Time

The partial representation extension problem generalizes the recognition problem for geometric intersection graphs. The input consists of a graph G, a subgraph \(H \subseteq G\) and a representation \(\mathcal R'\) of H. The question is whether G admits a representation \(\mathcal R\) whose restriction to H is \(\mathcal R'\). We study this question for circle graphs, which are intersection graphs of chords of a circle. Their representations are called chord diagrams. We show that for a graph with n vertices and m edges the partial representation extension problem can be solved in \(O((n + m) \alpha (n + m))\) time, thereby improving over an \(O(n^3)\)-time algorithm by Chaplick et al. (J Graph Theory 91(4), 365–394, 2019). The main technical contributions are a canonical way of orienting chord diagrams and a novel compact representation of the set of all canonically oriented chord diagrams that represent a given circle graph G, which is of independent interest.