Specifying the staircase kernel of a two-fold connected orthogonal polygon

Abstract

Let $$S\ne \phi $$ be a two-fold connected orthogonal polygon in the plane. Assume that S is starshaped via staircase paths and K is any component of $$\textit{Ker} S$$ , the staircase kernel of S. Let B be the bounded component of $$ \mathbb {R}^2{\setminus } S$$ . If B contains one kind of north, east, south and west dents then $$\textit{Ker} S$$ is either one component or two and the positions of these components agree with the kind of dents. But if B contains two kinds of north, east, south and west dents then either KerS is one component or S is not starshaped via staircase paths.