discrete structure

Discrete structures are the foundation for formulation of problems and algorithms. In the following paper, you will experience the use of discrete math for a thinning algorithm and its necessary and sufficient conditions for characterizing points. 

Paper “A parallel thinning algorithm for medial surfaces” G. Bertrand, Pattern Recognition Letters, 16 (1995) 979-986. This paper is attached. 

The overall idea of the algorithm is highlighted in blue, although the question to be answered is related to Proposition 2 (highlighted in yellow). Simple point is an important concept used for the algorithm. You are expected to answer the following question regarding a simple point mentioned in Proposition 2. 

Question 1. What is the definition of that is mentioned in Proposition 2? Briefly explain the meaning of it. Define each symbol used in your definition. 

Question 2. What is the definition of ? Briefly explain the meaning of it. 

Question 3. Proposition 2 say that x is n-simple if and only if   and . Give at least one example of x and X such that x is 6-simple. Briefly explain.