Extending Partial Representations of Circle Graphs in Near-Linear Time

Abstract

AbstractThe partial representation extension problem generalizes the recognition problem for geometric intersection graphs. The input consists of a graph G, a subgraph $$H \subseteq G$$ H ⊆ G and a representation $$\mathcal R'$$ R ′ of H. The question is whether G admits a representation $$\mathcal R$$ R whose restriction to H is $$\mathcal R'$$ 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 $$O((n + m) \alpha (n + m))$$ O ( ( n + m ) α ( n + m ) ) time, thereby improving over an $$O(n^3)$$ 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.