Vertex ordering and partitioning problems for random spatial graphs

@article{Penrose2000VertexOA,
  title={Vertex ordering and partitioning problems for random spatial graphs},
  author={Mathew D. Penrose},
  journal={Annals of Applied Probability},
  year={2000},
  volume={10},
  pages={517-538}
}
  • M. Penrose
  • Published 1 May 2000
  • Mathematics
  • Annals of Applied Probability
Given an ordering of the vertices of a finite graph, let the induced weight for an edge be the separation of its endpoints in the ordering. Layout problems involve choosing the ordering to minimize a cost functional such as the sum or maximum of the edge weights. We give growth rates for the costs of some of these problems on supercritical percolation processes and supercritical random geometric graphs, obtained by placing vertices randomly in the unit cube and joining them whenever at most some… 

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A convergence theorem for the optimal cost of the Minimum Linear Arrangement problem and the Minimum Sum Cut problem, for the case where the underlying graph is obtained through a subcritical site percolation process, can be viewed as an analogue of the Beardwood, Halton and Hammersley theorem forThe Euclidian TSP.

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