Burning Number for the Points in the Plane
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The burning process on a graph G starts with a single burnt vertex, and at each subsequent step, burns the neighbors of the currently burnt vertices, as well as one other unburnt vertex. The burning number of G is the smallest number of steps required to burn all the vertices of the graph. In this paper, we examine the problem of computing the burning number in a geometric setting. The input is a set of points P in the Euclidean plane. The burning process starts with a single burnt point, and at each subsequent step, burns all the points that are within a distance of one unit from the currently burnt points and one other unburnt point. The burning number of P is the smallest number of steps required to burn all the points of P. We call this variant point burning. We consider another variant called anywhere burning, where we are allowed to burn any point of the plane. We show that point burning and anywhere burning problems are both NP-complete, but (2+ε) approximable for every ε>0. Moreover, if we put a restriction on the number of burning sources that can be used, then the anywhere burning problem becomes NP-hard to approximate within a factor of 2/√(3)-ε.
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