A generalization of the Art Gallery Theorem

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. A classical result of Chvátal, the Art Gallery Theorem, tells us that the minimum size 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 ι k (G) denote the size of a smallest set S for which this is achieved. If n ≤ 2k + 3, then trivially ι k (G) ≤ 1. Let G be a mop on n ≥ max{5, 2k + 3} vertices, n 2 of which are of degree 2. Sharp bounds on ι k (G) have been obtained for k = 0 and k = 1, namely ι 0 (G) ≤ min{ n 4 , n+n2 5 , n−n2 3 } and ι 1 (G) ≤ min{ n 5 , n+n2 6 , n−n2 3 }. We prove that ι k (G) ≤ min{ n k+4 , n+n2 k+5 , n−n2 k+2 } for any k ≥ 0, and that this bound is sharp. We also prove that n−n2 2 is a sharp upper bound on the domination number of G.