Zero-visibility cops and robber and the pathwidth of a graph

We examine the zero-visibility cops and robber graph searching model, which differs from the classical cops and robber game in one way: the robber is invisible. We show that this model is not monotonic. We show that the zero-visibility copnumber of a graph is bounded above by its pathwidth and cannot be bounded below by any nontrivial function of the pathwidth. As well, we define a monotonic version of this game and show that the monotonic zero-visibility copnumber can be bounded both above and below by positive multiples of the pathwidth.

[1]  Volkan Isler,et al.  The role of information in the cop-robber game , 2008, Theor. Comput. Sci..

[2]  Boting Yang.  Strong-mixed searching and pathwidth , 2007, J. Comb. Optim..

[3]  Dirk Oliver Theis,et al.  The Cops and Robber game on graphs with forbidden (induced) subgraphs , 2010, Contributions Discret. Math..

[4]  Christos H. Papadimitriou,et al.  Interval graphs and seatching , 1985, Discret. Math..

[5]  T. D. Parsons,et al.  Pursuit-evasion in a graph , 1978 .

[6]  B. Alspach.  SEARCHING AND SWEEPING GRAPHS: A BRIEF SURVEY , 2006 .

[7]  Dimitrios M. Thilikos,et al.  An annotated bibliography on guaranteed graph searching , 2008, Theor. Comput. Sci..

[8]  J. Edmonds.  Paths, Trees, and Flowers , 1965, Canadian Journal of Mathematics.

[9]  Paul D. Seymour,et al.  Graph Minors: XV. Giant Steps , 1996, J. Comb. Theory, Ser. B.

[10]  Sampath Kannan,et al.  Randomized Pursuit-Evasion with Local Visibility , 2006, SIAM J. Discret. Math..

[11]  Anthony Bonato,et al.  The Game of Cops and Robbers on Graphs , 2011 .

[12]  Andrea S. LaPaugh,et al.  Recontamination does not help to search a graph , 1993, JACM.

[13]  Ivan Hal Sudborough,et al.  The Vertex Separation and Search Number of a Graph , 1994, Inf. Comput..

[14]  Paul D. Seymour,et al.  Graph minors. I. Excluding a forest , 1983, J. Comb. Theory, Ser. B.

[15]  Gary MacGillivray,et al.  Characterizations of k-copwin graphs , 2012, Discret. Math..

[16]  Athanasios Kehagias,et al.  Cops and invisible robbers: The cost of drunkenness , 2012, Theor. Comput. Sci..

[17]  Samuel Fiorini,et al.  A Note on the Cops and Robber Game on Graphs Embedded in Non-Orientable Surfaces , 2014, Graphs Comb..

[18]  Nancy G. Kinnersley,et al.  The Vertex Separation Number of a Graph equals its Path-Width , 1992, Inf. Process. Lett..

[19]  Boting Yang,et al.  Sweeping graphs with large clique number , 2009, Discret. Math..

[20]  Bernd S. W. Schröder.  The Copnumber of a Graph is Bounded by [3/2 genus ( G )] + 3 , 2001 .

[21]  Martin Aigner,et al.  A game of cops and robbers , 1984, Discret. Appl. Math..

[22]  Micah Adler,et al.  Randomized Pursuit-Evasion In Graphs , 2003, Comb. Probab. Comput..

[23]  Yuri Tschinkel,et al.  Trends in Mathematics , 2008 .

[24]  Dariusz Dereniowski.  From Pathwidth to Connected Pathwidth , 2012, SIAM J. Discret. Math..

[25]  Peter Winkler,et al.  Vertex-to-vertex pursuit in a graph , 1983, Discret. Math..

[26]  Athanasios Kehagias,et al.  Some remarks on cops and drunk robbers , 2011, Theor. Comput. Sci..

[27]  Paul D. Seymour,et al.  Monotonicity in Graph Searching , 1991, J. Algorithms.