Along with the direction that generalizes interval graphs and permutation graphs to trapezoid graphs, researchers are now trying to generalize the class known as trapezoid graphs. A circle trapezoid is the region in a circle that lies between two non-crossing chords; thus, circle trapezoid graphs are the intersecting graphs of circle trapezoids within a circle. It should be noted that circle tr...

Given a finite set of points P in R and a permutation of P , f : P → P , what is the minimum number of arcs required to connect the points of P such that every point p ∈ P is adjacent to f(p) along an arc and no two arcs cross? We show this question is NP-complete. 1 Problem Definition An arc consists of a closed continuous subset of a circle in the plane (the subset may be a complete circle). ...

Let G denote a graph and let K be a subset of vertices that are a set of target vertices of G. The Kterminal reliability of G is defined as the probability that all target vertices in K are connected, considering the possible failures of non-target vertices of G. The problem of computing K-terminal reliability is known to be #P-complete for polygon-circle graphs, and can be solved in polynomial...

An essential part of visual object recognition is the evaluation of the curvature of both an object's outline as well as the contours on its surface. We studied a striking illusion of visual curvature--the arc-size illusion (ASI)--to gain insight into the visual coding of curvature. In the ASI, short arcs are perceived as flatter (less curved) compared to longer arcs of the same radius. We inve...

We study geometric properties of horopters defined by the criterion of equality of angle. Our primary goal is to derive the precise geometry for anatomically correct horopters. When eyes fixate on points along a curve in the horizontal visual plane for which the vergence remains constant, this curve is the larger arc of a circle connecting the eyes' rotation centers. This isovergence circle is ...

A circular-arc model (C,A) is a circle C together with a collection A of arcs of C. If A satisfies the Helly Property then (C,A) is a Helly circular-arc model. A (Helly) circular-arc graph is the intersection graph of a (Helly) circular-arc model. Circular-arc graphs and their subclasses have been the object of a great deal of attention, in the literature. Linear time recognition algorithm have...

The circular arc coloring problem is to find a minimum coloring of a set of arcs of a circle so that no two overlapping arcs share a color. This N P-hard problem arises in a rich variety of applications and has been studied extensively. In this paper we present an O(n2m) combinatorial algorithm for optimally coloring any set of arcs that corresponds to a perfect graph, and propose a new approac...

Circular graphs are intersection graphs of arcs on a circle. These graphs are reported to have been studied since 1964, and they have been receiving considerable attention since a series of papers by Tucker in the 1970s. Various subclasses of circular-arc graphs have also been studied. Among these are the proper circular-arc graphs, unit circular-arc graphs, Helly circular-arc graphs and co-bip...

An arc-representation of a graph is a function mapping each vertex in the graph to an arc on the unit circle in such a way that adjacent vertices are mapped to intersecting arcs. The width of such a representation is the maximum number of arcs passing through a single point. The arc-width of a graph is defined to be the minimum width over all of its arc-representations. We extend the work of Ba...

We prove that the Lipschitz dimension of any bounded turning Jordan circle or arc is equal to 1. Equivalently, weak quasicircle