Extensions of the Art Gallery Theorem

TL;DR

This paper extends the Art Gallery Theorem by providing new bounds on the number of guards needed in polygons, considering maximum degree constraints and degree-two vertices, improving previous bounds and proving their sharpness.

Contribution

It introduces generalized bounds for domination in maximal outerplanar graphs, extending the classical Art Gallery Theorem with new parameters and proving their optimality.

Findings

New upper bounds for domination numbers in maximal outerplanar graphs.

Extension of the Art Gallery Theorem to scenarios with degree constraints.

Proof that the bounds are sharp and improvements over previous results.

Abstract

Several domination results have been obtained for maximal outerplanar graphs (mops). The classical domination problem is to minimize the size of a set $S$ of vertices of an $n$-vertex graph $G$ such that $G - N[S]$, the graph obtained by deleting the closed neighborhood of $S$, contains no vertices. In the proof of the Art Gallery Theorem, Chv\'{a}tal showed that the minimum size, called the domination number of $G$ and denoted by $\gamma(G)$, is at most $n/3$ if $G$ is a mop. Here we consider a modification by allowing $G - N[S]$ to have a maximum degree of at most $k$. Let $\iota_k(G)$ denote the size of a smallest set $S$ for which this is achieved. If $n \le 2k+3$, then trivially $\iota_k(G) \leq 1$. Let $G$ be a mop on $n \ge \max\{5,2k+3\}$ vertices, $n_2$ of which are of degree $2$. Upper bounds on $\iota_k(G)$ have been obtained for $k = 0$ and $k = 1$, namely $\iota_{0}(G) \le \min\{\frac{n}{4},\frac{n+n_2}{5},\frac{n-n_2}{3}\}$ and $\iota_1(G) \le \min\{\frac{n}{5},\frac{n+n_2}{6},\frac{n-n_2}{3}\}$. We prove that $\iota_{k}(G) \le \min\{\frac{n}{k+4},\frac{n+n_2}{k+5},\frac{n-n_2}{k+2}\}$ for any $k \ge 0$. For the original setting of the Art Gallery Theorem, the argument presented yields that if an art gallery has exactly $n$ corners and at least one of every $k + 2$ consecutive corners must be visible to at least one guard, then the number of guards needed is at most $n/(k+4)$. We also prove that $\gamma(G) \le \frac{n - n_2}{2}$ unless $n = 2n_2$, $n_2$ is odd, and $\gamma(G) = \frac{n - n_2 + 1}{2}$. Together with the inequality $\gamma(G) \le \frac{n+n_2}{4}$, obtained by Campos and Wakabayashi and independently by Tokunaga, this improves Chv\'{a}tal's bound. The bounds are sharp.