Krasnosel'skii numbers and non simply connected orthogonal polygons. (Q2715987)

An orthogonal (rectilinear) polygon is a region in the plane bounded by finitely many segments, each parallel to a coordinate axis. A staircase path consists of a finite number of axis-parallel segments and it is monotone, meaning that when following the path from one end to the other, we never travel in two parallel but opposite directions. Points \(x,y\) in an orthogonal polygon \(P\) are staircase visible if they can be connected by a staircase path within \(P\). It was known that the Krasnosel'skii number for staircase visibility in simply connected orthogonal polygons is 2, meaning that if every two points of the boundary of \(P\) are staircase visible from some point of \(P\) then there is a point from which all of \(P\) is staircase visible. The paper shows that the Krasnosel'skii number of a polygon obtained by deleting \(n\) axis-parallel rectangles from a simply connected orthogonal polygon is at most \(4n\), and can be at least const.\, \(n\).