Abstract: The intersection graph of a set of chords on a circle is called a circle graph. This class of graphs admits some interesting subclasses, such as: Helly circle graphs, clique-Helly circle graphs, unit circle graphs and proper circular-arc graphs. In this paper, we prove some inclusion relations among these subclasses. A necessary condition for a graph being a Helly circle graph is show...

Polygon-circle graphs are intersection graphs of polygons inscribed in a circle. This class of graphs includes circle graphs (intersection graphs of chords of a circle), circular arc graphs (intersection graphs of arcs on a circle), chordal graphs and outerplanar graphs. We investigate binding functions for chromatic number and clique covering number of polygon-circle graphs in terms of their c...

We prove an interpolation theorem for rational circle diffeomorphisms: A set of N complex numbers of unit modulus may be mapped to any corresponding set by a ratio of polynomials p(z)/q(z) which restricts an invertible mapping of the unit circle S ⊂ C onto itself.

All the mouse embryos above belong to the same litter but half of them have inherited a transgene from their father (the green fluorescent adult mouse on the right) that causes them to glow yellowish green under fluorescent light. In these transgenic mice, the enhanced green fluorescent protein (EGFP) is expressed in all cells except hair and erythrocytes. The fact that there are equal numbers ...

Let d > 2 be an integer number, and let fk, k ∈ {1, . . . , d}, be Ck commuting circle diffeomorphisms, with τk ∈ ]0, 1[ and τ1 + · · · + τd > 1. We prove that if the rotation numbers of the fk’s are independent over the rationals (that is, if the corresponding action of Z n on the circle is free), then they are simultaneously (topologically) conjugate to rotations. 2000 Math. Subj. Class. 22F0...

A circle graph is the intersection graph of a set of chords of a circle. The class of circle graphs is closed under pivotminors. We determine the pivot-minor-minimal non-circle-graphs; there 15 obstructions. These obstructions are found, by computer search, as a corollary to Bouchet’s characterization of circle graphs under local complementation. Our characterization generalizes Kuratowski’s Th...