Extending Partial Representations of Circle Graphs

TL;DR

This paper presents an $O(n^3)$ algorithm for extending partial circle graph representations, utilizing split decomposition to understand all possible representations, advancing the understanding of recognition and extension problems.

Contribution

The paper introduces a polynomial-time algorithm for extending partial representations of circle graphs, based on a novel structural analysis via split decomposition.

Findings

Developed an $O(n^3)$ algorithm for partial representation extension.

Characterized all circle graph representations using split decomposition.

Provides a new structural insight into circle graph representations.

Abstract

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 $\cal R'$ giving some pre-drawn chords that represent an induced subgraph of $G$. The question is whether one can extend $\cal R'$ to a representation $\cal R$ of the entire graph $G$, i.e., whether one can draw the remaining chords into a partially pre-drawn representation to obtain a representation of $G$. Our main result is an $O(n^3)$ time algorithm for partial representation extension of circle graphs, where $n$ 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.