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

  • Authors:

  • David Auger;Irène Charon;Olivier Hudry;Antoine Lobstein

  • Affiliations:

  • Institut Télécom - Télécom ParisTech & Centre National de la Recherche Scientifique - LTCI UMR 5141, 46, rue Barrault, 75634 Paris Cedex 13, France;Institut Télécom - Télécom ParisTech & Centre National de la Recherche Scientifique - LTCI UMR 5141, 46, rue Barrault, 75634 Paris Cedex 13, France;Institut Télécom - Télécom ParisTech & Centre National de la Recherche Scientifique - LTCI UMR 5141, 46, rue Barrault, 75634 Paris Cedex 13, France;Centre National de la Recherche Scientifique - LTCI UMR 5141 & Institut Téélécom - Télécom ParisTech, 46, rue Barrault, 75634 Paris Cedex 13, France

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2014

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Abstract

Let G=(V(G),E(G)) be an undirected graph. A watcher w of G is a couple w = (@?(w), A(w)), where @?(w) belongs to V(G) and A(w) is a set of vertices of G at distance 0 or 1 from @?(w). 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 v"2. 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@?W, and every two vertices v"1,v"2 are separated by W. The minimum number of watchers necessary to watch G is denoted by w(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.