January 2025
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3 Reads
Discrete Mathematics
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January 2025
·
3 Reads
Discrete Mathematics
September 2024
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7 Reads
We investigate a cheating robot version of Cops and Robber, first introduced by Huggan and Nowakowski, where both the cops and the robber move simultaneously, but the robber is allowed to react to the cops' moves. For conciseness, we refer to this game as Cops and Cheating Robot. The cheating robot number for a graph is the fewest number of cops needed to win on the graph. We introduce a new parameter for this variation, called the push number, which gives the value for the minimum number of cops that move onto the robber's vertex given that there are a cheating robot number of cops on the graph. After producing some elementary results on the push number, we use it to give a relationship between Cops and Cheating Robot and Surrounding Cops and Robbers. We investigate the cheating robot number for planar graphs and give a tight bound for bipartite planar graphs. We show that determining whether a graph has a cheating robot number at most fixed k can be done in polynomial time. We also obtain bounds on the cheating robot number for strong and lexicographic products of graphs.
August 2024
We introduce the bodyguard problem for graphs. This is a variation of Surrounding Cops and Robber but, in this model, a smallest possible group of bodyguards must surround the president and then maintain this protection indefinitely. We investigate some elementary bounds, then solve this problem for the infinite graph families of complete graphs, wheels, trees, cycles, complete multipartite graphs, and two-dimensional grids. We also examine the problem in more general Cartesian, strong, and lexicographic products.
November 2022
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38 Reads
Graph burning is a discrete-time process that models the spread of influence in a network. Vertices are either burning or unburned, and in each round, a burning vertex causes all of its neighbours to become burning before a new fire source is chosen to become burning. We introduce a variation of this process that incorporates an adversarial game played on a nested, growing sequence of graphs. Two players, Arsonist and Builder, play in turns: Builder adds a certain number of new unburned vertices and edges incident to these to create a larger graph, then every vertex neighbouring a burning vertex becomes burning, and finally Arsonist `burns' a new fire source. This process repeats forever. Arsonist is said to win if the limiting fraction of burned vertices tends to 1, while Builder is said to win if this fraction is bounded away from 1. The central question of this paper is determining if, given that Builder adds f(n) vertices at turn n, either Arsonist or Builder has a winning strategy. In the case that f(n) is asymptotically polynomial, we give a threshold result for which player has a winning strategy.