Finding a maximum set of independent chords in a circle

Abstract Chang, R.C. and H.S. Lee, Finding a maximum set of independent chords in a circle, Information Processing Letters 41 (1992) 99-102. In this note we propose an O(nmI algorithm for finding a maximum independent set of m chords which are incident to n vertices on a circle. This result can be applied to improving the time complexity of the algorithm for partitioning simple polygons into a minimum number of uniformly monotone polygons

Finding a Maximum Weight Independent Set of a Circle Graph

We present an algorithm for finding a maximum weight independent set of a circle graph. For a cicle graph of a set of n chords with N endpoints, the algorithm finds a maximum weight independent set in O(nN) time and O(n) space.Special Section LETTER (Special Issue on Discrete Mathematics and Its Applications

Minimizing Crossings in Constrained Two-Sided Circular Graph Layouts

Circular layouts are a popular graph drawing style, where vertices are placed on a circle and edges are drawn as straight chords. Crossing minimization in circular layouts is NP-hard. One way to allow for fewer crossings in practice are two-sided layouts that draw some edges as curves in the exterior of the circle. In fact, one- and two-sided circular layouts are equivalent to one-page and two-page book drawings, i.e., graph layouts with all vertices placed on a line (the spine) and edges drawn in one or two distinct half-planes (the pages) bounded by the spine. In this paper we study the problem of minimizing the crossings for a fixed cyclic vertex order by computing an optimal k-plane set of exteriorly drawn edges for k >= 1, extending the previously studied case k=0. We show that this relates to finding bounded-degree maximum-weight induced subgraphs of circle graphs, which is a graph-theoretic problem of independent interest. We show NP-hardness for arbitrary k, present an efficient algorithm for k=1, and generalize it to an explicit XP-time algorithm for any fixed k. For the practically interesting case k=1 we implemented our algorithm and present experimental results that confirm the applicability of our algorithm