
Adjacency Labelling for Planar Graphs (and Beyond)
We show that there exists an adjacency labelling scheme for planar graph...
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The Pyro game: a slow intelligent fire
In the Firefighter problem, a fire breaks out at a vertex of a graph and...
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Shorter Labeling Schemes for Planar Graphs
An adjacency labeling scheme for a given class of graphs is an algorithm...
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On Adjacency and eadjacency in General Hypergraphs: Towards an eadjacency Tensor
Adjacency between two vertices in graphs or hypergraphs is a pairwise re...
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On Adjacency and eAdjacency in General Hypergraphs: Towards a New eAdjacency Tensor
In graphs, the concept of adjacency is clearly defined: it is a pairwise...
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Complexity of the Multilevel Critical Node Problem
In this work, we analyze a sequential game played in a graph called the ...
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Firefighting on Trees
In the Firefighter problem, introduced by Hartnell in 1995, a fire sprea...
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Multilayered planar firefighting
Consider a model of fire spreading through a graph; initially some vertices are burning, and at every given timestep fire spreads from burning vertices to their neighbours. The firefighter problem is a solitaire game in which a player is allowed, at every timestep, to protect some nonburning vertices (by effectively deleting them) in order to contain the fire growth. How many vertices per turn, on average, must be protected in order to stop the fire from spreading infinitely? Here we consider the problem on ℤ^2× [h] for both nearest neighbour adjacency and strong adjacency. We determine the critical protection rates for these graphs to be 1.5h and 3h, respectively. This establishes the fact that using an optimal twodimensional strategy for all layers in parallel is asymptotically optimal.
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