This work deals with convergence theorems and bounds on the cost of several layout
measures for lattice graphs, random lattice graphs and sparse random geometric graphs.
Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth,
Sum Cut, Vertex Separation, Edge Bisection and Vertex Bisection. For full square lattices,
we give optimal layouts for the problems still open. For arbitrary lattice graphs, we present
best possible bounds disregarding a constant factor. We apply percolation theory to the
study of lattice graphs in a probabilistic setting. In particular, we deal with the subcritical
regime that this class of graphs exhibits and characterize the behaviour of several layout
measures in this space of probability. We extend the results on random lattice graphs to
random geometric graphs, which are graphs whose nodes are spread at random in the unit
square and whose edges connect pairs of points which are within a given distance. We also
characterize the behaviour of several layout measures on random geometric graphs in their
subcritical regime. Our main results are convergence theorems that can be viewed as an
analogue of the Beardwood, Halton and Hammersley theorem for the Euclidean TSP on
random points in the unit square.
Convergence Theorems for Some Layout Measures on Random Lattice and Random Geometric Graphs