Extensions of the Art Gallery Theorem

Several domination results have been obtained for maximal outerplanar graphs (mops). The classical problem is to minimize the size of a set S vertices an n-vertex graph G such that $$G - N[S]$$ , by deleting closed neighborhood S, contains no vertices. In proof Art Gallery Theorem, Chvátal showed minimum size, called number and denoted $$\gamma (G)$$ at most n/3 if mop. Here we consider modification allowing maximum degree k. Let $$\iota _k(G)$$ denote smallest which this achieved. If $$n \le 2k+3$$ then trivially _k(G) 1$$ . be mop on \ge \max \{5,2k+3\}$$ vertices, $$n_2$$ are 2. Upper bounds $$k = 0$$ namely _{0}(G) \min \{\frac{n}{4},\frac{n+n_2}{5},\frac{n-n_2}{3}\}$$ _1(G) \{\frac{n}{5},\frac{n+n_2}{6},\frac{n-n_2}{3}\}$$ We prove _{k}(G) \{\frac{n}{k+4},\frac{n+n_2}{k+5},\frac{n-n_2}{k+2}\}$$ any For original setting argument presented yields art gallery has exactly n corners least one every + 2$$ consecutive must visible guard, guards needed $$n/(k+4)$$ also (G) \frac{n n_2}{2}$$ unless 2n_2$$ odd, n_2 1}{2}$$ Together with inequality \frac{n+n_2}{4}$$ Campos Wakabayashi independently Tokunaga, improves Chvátal’s bound. sharp.