The relation between the staircase kernel and the position of the three staircase paths of the bounded component for the complement

One of the most important subjects that the starshaped sets theory concerned withis specifying the kernel of the starshaped set and vision the points and regions for each other. So in staircase visibilitytheresearcher Rajeev Motwani proved that the points of separating regions with dents cannot see each other. After that Breen could find a way for specifying the kernel of starshaped orthogonal polygon when this orthogonal polygon is simply connected. In this paper we will generalize the previous way when the closed orthogonal polygon is secondly connected and the bounded component for the complement is union of three staircase paths, every path consists of more than two edges. We will prove that the kernel is only one component.

The Influence of the Bounded Component for the Complement in a Starshaped Orthogonal Polygon on its Kernel

Many mathematicians were interested in specifying the kernel of the starshaped set. In staircase visibility the researcher Rajeev Motwani proved that the points of separating regions with dents cannot see each other, and then he proved that these points are seen from other points of an orthogonal polygon. After that Breen could find a way for specifying the kernel of starshaped orthogonal polygon when this orthogonal polygon is simply connected. The aim of this paper is to generalize the previous way when the closed orthogonal polygon is secondly connected and the bounded component for the complement contains one staircase path or two staircase paths, every path consists of more than two edges. We will prove that the kernel is either one component or two or three ones.

Specifying the Kernels of the Starshaped Orthogonal Polygons which are Secondly Connected when the Component of the Complement is a Rectangle

The staircase visibility concerns with the study of orthogonal polygon, one of the most important subjects which are studied is the Specification kernel of the orthogonal starshaped set. Toranzos represent a very important result in Specifying the kernel of the starshaped set in the usual notion of visibility via segments, after that Breen presented an analogue to this result of the staircase visibility. She also could find a way for Specifying the kernel of starshaped orthogonal polygon when this orthogonal polygon is a simply connected. The aim of this paper is generalizing the previous way when the orthogonal polygon is secondly connected and the bounded component for the complement is a rectangular; we will prove the following result: Let , be secondly connected closed orthogonal polygon, and staircase starshaped set. If the boundary of the bounded component for the complement is a rectangle ,so the kernel of is either one component or two or four ones.