How to Guard Orthogonal Polygons: Diagonal Graphs and Vertex Covers

Abstract

We extend and unify most known results about guarding orthogonal polygons by introducing the same-sign diagonal graphs of a convex quadrangulation and applying results about vertex covers for graphs. Our approach also yields new theorems and often guarantees two disjoint vertex guard sets of relatively small cardinality. For instance, an orthogonal polygon on n vertices has two disjoint vertex guard sets of cardinality at most $\lfloor n/4\rfloor$. We give new proofs of Aggarwal’s one-hole theorem and the orthogonal fortress theorem. We prove that an orthogonal polygon with n vertices and any number of holes can be protected by at most $\lfloor (17n-8)/52\rfloor$ vertex guards, improving the best known bound of $\lfloor n/3\rfloor$. Also, an orthogonal polygon with n vertices and h holes can be protected by $\lfloor (n+2h)/3\rfloor$ guarded guards, which is best possible when $n\ge 16h$. Moreover, for orthogonal fortresses with n vertices, $\lfloor (n+6)/3\rfloor$ guarded guards are always sufficient and sometimes necessary.