We prove that a graph is 4-colorable if and only if it can be drawn with vertices in the integer lattice, using as edges only line segments not containing a third point of the lattice.

We consider visibility representations of graphs in which the vertices are represented by a collection O of non-overlapping convex regions on the plane. Two points x and y are visible if the straight-line segment xy is not obstructed by any object. Two objects A; B 2 O are called visible if there exist points x 2 A; y 2 B such that x is visible from y. We consider visibility only for a nite set...

atmospheric visibility is a key factor in everyday life mainly in aviation, industry and surface traffic. it has been defined as the greatest distance at which an observer can just see a black object viewed against the horizon sky which is known as the visual range. furthermore, light extinction which is mostly due to absorption and scattering effects of aerosols in the atmosphere, can be calcu...

We show that every segment endpoint visibility graph on disjoint line segments in the plane admits an alternating path of length , answering a question of Bose. This bound is optimal apart from a constant factor. We also give bounds on the constants hidden by the asymptotic notation.

Visibility representation of digraphs was introduced by Axenovich, Beveridge, Hutch\-inson, and West (\emph{SIAM J. Discrete Math.} {\bf 27}(3) (2013) 1429--1449) as a natural generalization $t$-bar visibility undirected graphs. A {\it representation} digraph $G$ assigns each vertex at most $t$ horizontal bars in the plane so that there is an arc $xy$ if only some bar for $x$ "sees" $y$ above i...

In the first part of this survey we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the rectangular dual. In visibility graphs and segment contact graphs the vertices correspond to horizonta...

INTRODUCTION A variety of mesial and distal projections have been suggested for separating the canals in a multi-canaled root. But there is no general agreement on the best angulation for each tooth. The purpose of this study was to evaluate the X-ray horizontal angle for separating the canals of four-canal first mandibular molars. MATERIALS AND METHODS Forty four-canaled mandibular molars we...

In the present study we investigated steering accuracy in terms of our ability to keep to the middle of a lane in a xed-base driving simulator. In particular, we studied the dependence of steering accuracy on the visibility o f diierent road sections, on the assumption that performance reeects the importance of diierent road sections in guiding steering. Other innuences on steering accuracy-inc...