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Creators/Authors contains: "Dudek, A"

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  1. ; ; ; ; (, Discrete applied mathematics)
    We consider the localization game played on graphs in which a cop tries to determine the exact location of an invisible robber by exploiting distance probes. The corresponding graph parameter $$\zeta(G)$$ for a given graph $$G$$ is called the localization number. In this paper, we improve the bounds for dense random graphs determining an asymptotic behaviour of $$\zeta(G)$$. Moreover, we extend the argument to sparse graphs. 
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    Accepted Manuscript Full Text Available
  2. ; ; (, SIAM journal on discrete mathematics)
    The game of plates and olives was originally formulated by Nicolaescu and encodes the evolution of the topology of the sublevel sets of Morse functions. We consider a random variant of this game. The process starts with an empty table. There are four different types of moves: (1) add a new plate to the table, (2) combine two plates and their olives onto one plate, removing the second plate from the table, (3) add an olive to a plate, and (4) remove an olive from a plate. We show that with high probability the number of olives is linear as the total number of moves goes to infinity. Furthermore, we prove that the number of olives is concentrated around its expectation. 
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    Accepted Manuscript Full Text Available