DeepAI

Pursuit-Evasion in Graphs: Zombies, Lazy Zombies and a Survivor

We study zombies and survivor, a variant of the game of cops and robber on graphs. In this variant, the single survivor plays the role of the robber and attempts to escape from the zombies that play the role of the cops. The zombies are restricted, on their turn, to always follow an edge of a shortest path towards the survivor. Let z(G) be the smallest number of zombies required to catch the survivor on a graph G with n vertices. We show that there exist outerplanar graphs and visibility graphs of simple polygons such that z(G) = Θ(n). We also show that there exist maximum-degree-3 outerplanar graphs such that z(G) = Ω(n/log(n)). Let z_L(G) be the smallest number of lazy zombies (zombies that can stay still on their turn) required to catch the survivor on a graph G. We establish that lazy zombies are more powerful than normal zombies but less powerful than cops. We prove that z_L(G) = 2 for connected outerplanar graphs. We show that z_L(G)≤ k for connected graphs with treedepth k. This result implies that z_L(G) is at most (k+1)log n for connected graphs with treewidth k, O(√(n)) for connected planar graphs, O(√(gn)) for connected graphs with genus g and O(h√(hn)) for connected graphs with any excluded h-vertex minor. Our results on lazy zombies still hold when an adversary chooses the initial positions of the zombies.

• 33 publications
• 13 publications
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12/12/2017

Minors of two-connected graphs of large path-width

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Cops and robber on subclasses of P_5-free graphs

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Isolation of connected graphs

For a connected n-vertex graph G and a set ℱ of graphs, let ι(G,ℱ) denot...
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Connected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly

The Grundy number of a graph is the maximum number of colours used by th...
04/02/2020

A Spectral Approach to the Shortest Path Problem

Let G=(V,E) be a simple, connected graph. One is often interested in a s...
06/16/2020

Confining the Robber on Cographs

In this paper, the notions of trapping and confining the robber on a gra...
10/31/2017

Variations of the cop and robber game on graphs

We prove new theoretical results about several variations of the cop and...