Maximum size of a minimum watching system and the graphs achieving the bound

Let G = (V (G), E(G)) be an undirected graph. A watcher to of G is a couple w = (E(w), A(w)), where t(w) belongs to V (G) and A(w) is a set of vertices of G at distance 0 or 1 from aw). If a vertex v belongs to A(w), we say that v is covered by w. Two vertices v(1) and v(2) in G are said to be separated by a set of watchers if the list of the watchers covering v(1) is different from that of 02. We say that a set W of watchers is a watching system for G if every vertex v is covered by at least one w E W, and every two vertices v(1), 02 are separated by W. The minimum number of watchers necessary to watch G is denoted by to (G). We give an upper bound on w(G) for connected graphs of order n and characterize the trees attaining this bound, before studying the more complicated characterization of the connected graphs attaining this bound. (C) 2012 Elsevier B.V. All rights reserved.