Ortho-unit polygons can be guarded with at most $\lfloor \frac{n-4}{8} \rfloor$ guards

TL;DR

This paper proves a tight upper bound on the number of guards needed to cover any ortho-unit polygon with at least 12 vertices, based on its vertex count.

Contribution

It establishes a precise guard number bound for ortho-unit polygons, extending guard theory to a specific class of polygons with integer-coordinate edges.

Findings

Any ortho-unit polygon with n ≥ 12 vertices can be guarded with at most ⌊(n-4)/8⌋ guards.

The bound ⌊(n-4)/8⌋ is proven to be tight.

The result applies to polygons with integer vertices and unit-length edges.

Abstract

An orthogonal polygon is called an ortho-unit polygon if its vertices have integer coordinates, and all of its edges have length one. In this paper we prove that any ortho-unit polygon with $n \geq 12$ vertices can be guarded with at most $\lfloor \frac{n-4}{8} \rfloor$ guards, which is a tight bound.