Convergence theorems for some layout measures on random lattice and random geometric graphs * Josep D´ ıaz † Mathew D. Penrose ‡ Jordi Petit † Mar´ ıa Serna † Abstract This work deals with convergence theorems and bounds on the cost of several lay- out measures for lattice graphs, random lattice graphs and sparse random geometric graphs. Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth, Sum Cut, V ertex Separation, Edge Bisection and V ertex 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 perco- lation 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 behavior 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 behavior of several layout measures on random geometric graphs in their subcritical regime. Our main results are convergence theorems that can be viewed as an analog of the Beardwood, Halton and Hammersley theorem for the Euclidean TSP on random points in the unit square. * This research was partially supported by ESPRIT LTR Project no. 20244 — ALCOM-IT, CICYT Project TIC97-1475-CE, and CIRIT project 1997SGR-00366. † Departament de Llenguatges i Sistemes Inform` atics. Universitat Polit` ecnica de Catalunya. Campus Nord C6. c/ Jordi Girona 1-3. 08034 Barcelona (Spain). {diaz,jpetit,mjserna}@lsi.upc.es ‡ Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, England. Mathew.Penrose@durham.ac.uk 1