A witness set W in a polygon P is a subset of P such that any set $G\subset P$ that guards W is guaranteed to guard P. We study the problem of finding a minimum witness set for an orthogonal polygon under three models of orthogonal visibility.

It is known that not all simple polygons admit a finite witness set under the traditional line-segment visibility and, if a polygon admits a finite minimal witness set, then the witnesses must lie on the boundary of the polygon [5]. In this paper, we prove that every orthogonal polygon with n vertices admits a finite witness set with $O(n^2)$ witnesses under rectangular, staircase, and k-periscope visibility. This upper bound is tight under staircase visibility. We also show an orthogonal polygon whose boundary is not a witness set for any of the three considered visibility models. We finally describe how to find a minimum witness set for a given orthogonal polygon in $O(n^4)$ time under rectangular and staircase visibility, and in $O(n^6)$ time under k-periscope visibility.

甲证人集合W中的多边形P的一个子集P，使得任何一组$G\subset P$卫兵W¯¯是保证后卫P。我们研究了在三个正交可见性模型下为正交多边形找到最小见证集的问题。

众所周知，并非所有简单的多边形都在传统的线段可见性下接受有限的见证集，并且，如果多边形接受有限的最小见证集，则见证人必须位于多边形的边界上[5]。在本文中，我们证明了每个具有n个顶点的正交多边形都可以通过$Ø（{ñ}^2）$矩形，阶梯和k潜望镜下的目击者。在楼梯的能见度下，这个上限很紧。我们还显示了一个正交多边形，其边界不是三个考虑的可见性模型中任何一个的见证集。最后，我们描述了如何在给定的正交多边形中找到最小见证集。$Ø（{ñ}^4）$ 在矩形和楼梯能见度下的时间，以及 $Ø（{ñ}^6）$k潜望镜可见性下的时间。