In random geometric graphs, vertices are randomly distributed on [0,1]^2 and pairs of vertices are connected by edges
whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a
certain measure is minimized. In this paper, we study several layout problems on random geometric graphs: Bandwidth,
Minimum Linear Arrangement, Minimum Cut, Minimum Sum Cut, Vertex Separation and Bisection. We first prove that
some of these problems remain...
In random geometric graphs, vertices are randomly distributed on [0,1]^2 and pairs of vertices are connected by edges
whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a
certain measure is minimized. In this paper, we study several layout problems on random geometric graphs: Bandwidth,
Minimum Linear Arrangement, Minimum Cut, Minimum Sum Cut, Vertex Separation and Bisection. We first prove that
some of these problems remain \NP-complete even for geometric graphs. Afterwards, we compute lower bounds that hold
with high probability on random geometric graphs. Finally, we characterize the probabilistic behavior of the lexicographic
ordering for our layout problems on the class of random geometric graphs.
Citation
Diaz, J., Penrose, M., Petit, J., Serna, M. "Linear orderings of random geometric graphs (extended abstract)". 1999.