We present an output sensitive algorithm for computing a maximum independent set of an unweighted circle graph. Our algorithm requires O(nmin{d, α}) time at worst, for an n vertex circle graph where α is the independence number of the circle graph and d is its density. Previous algorithms for this problem required Θ(nd) time at worst.

We study the complexity of the colouring problem for circle graphs. We will solve the two open questions of [Un88], where first results were presented. 1. Here we will present an algorithm which solves the 3-colouring problem of circle graphs in time O(n log(n)). In [Un88] we showed that the 4-colouring problem for circle graphs is NP-complete. 2. If the largest clique of a circle graph has siz...

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 R′ giving some pre-drawn chords that represent an induced subgraph of G. The ques...

A circle in a graph G is a homeomorphic image of the unit circle in the Freudenthal compactification of G, a topological space formed from G and the ends of G. Bruhn conjectured that every locally finite 4-connected planar graph G admits a Hamilton circle, a circle containing all points in the Freudenthal compactification of G that are vertices and ends of G. We prove this conjecture for graphs...

The Koebe-Andreev-Thurston Circle Packing Theorem states that every triangulated planar graph has a circle-contact representation. The theorem has been generalized in various ways. The arguably most prominent generalization assures the existence of a primal-dual circle representation for every 3-connected planar graph. The aim of this note is to give a streamlined proof of this result.

From the four considered algorithms the Minimum Spanning Tree (MST) is the only one resulting in a bounded-degree topology. Figure 1 depicts worst-case instances for the different graphs. Figure 1(a) gives thereby an idea why the degree of the MST is bounded to six. If the angle between two adjacent edges (u, v) and (u,w) is less than 60 degrees it would be cheaper to abandon one of these edges...