Partitioning Orthogonal Polygons by Extension of All Edges Incident to Reflex Vertices: Lower and Upper Bounds on the Number of Pieces

Abstract

Given an orthogonal polygon P , let |Π(P )| be the number of rectangles that result when we partition P by extending the edges incident to reflex vertices towards INT(P ). In [4] we showed that |Π(P )| ≤ 1 + r + r 2 , where r is the number of reflex vertices of P . We shall now give sharper bounds both for maxP |Π(P )| and minP |Π(P )|. Moreover, we characterize the structure of orthogonal polygons in general position for which these new bounds are exact.

Characterizing and Covering Some Subclasses of Orthogonal Polygons

“…, x = n 2 and that its northwest corner is (1, 1). Grid n-ogons that are symmetrically equivalent are grouped in the same class [1]. A grid n-ogon Q is called Fat iff |Π(Q)| ≥ |Π(P )|, for all grid n-ogons P .…”

“…Similarly, a grid n-ogon Q is called Thin iff |Π(Q)| ≤ |Π(P )|, for all grid n-ogons P . Let P be a grid n-ogon with r reflex vertices, in [1] is proved that, if P is Fat then |Π(P )| = 3r 2 +6r+4 4 , for r even and |Π(P…”

“…1 The area of a grid n-ogon is the number of grid cells in its interior. In [1] it is proved that for all grid n-ogon P , with n ≥ 8, 2r + 1 ≤ A(P ) ≤ r 2 + 3. A grid n-ogon P is a Max-Area grid n-ogon iff A(P ) = r 2 + 3 and it is a Min-Area grid n-ogon iff A(P ) = 2r + 1.…”