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**1 - 3**of**3**### Salient and Reentrant Points of Discrete Sets

"... The border-salient and reentrant points of a discrete set are special points of the border of the set. When they are given with multiplicity they completely characterize the set, and without multiplicity they characterize the set if all its 8-components are 4-connected. The inner-salient and reentra ..."

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The border-salient and reentrant points of a discrete set are special points of the border of the set. When they are given with multiplicity they completely characterize the set, and without multiplicity they characterize the set if all its 8-components are 4-connected. The inner-salient and reentrant are defined similarly to the border ones, but we show that, in general, they do not characterize the set, even if this set is 4-simply connected. We also show that the genus of a set can be easily computed from the number of salient and reentrant points. A discrete set is a finite subset of the integer plane Z 2. Intuitively a discrete set can be described by its border, but this border can also be characterized by the points where there is a change of direction. In this paper these points are said to be salient and reentrant points. In fact we define two types of salient and reentrant points: the first ones are on the border of the set (the border points), and the second ones are in the inner of the set (the inner points). Section 1 of this paper presents some preliminary definitions and properties.

### Random generation of Q-convex sets

"... The problem of randomly generating Q-convex sets is considered. We present two generators. The first one uses the Q-convex hull of a set of random points in order to generate a Q-convex set included in the square [0, n) 2. This generator is very simple, but is not uniform and its performance is quad ..."

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The problem of randomly generating Q-convex sets is considered. We present two generators. The first one uses the Q-convex hull of a set of random points in order to generate a Q-convex set included in the square [0, n) 2. This generator is very simple, but is not uniform and its performance is quadratic in n. The second one exploits a coding of the salient points, which generalizes the coding of a border of polyominoes. It is uniform, and is based on the method by rejection. Experimentally, this algorithm works in linear time in the length of the word coding the salient points. Besides, concerning the enumeration problem, we determine an asymptotic formula for the number of Q-convex sets according to the size of the word coding the salient points in a special case, and in general only an experimental estimation. Key words: uniform generator, lattice sets, convexity, salient points 1