# Linear Orderings of Random Geometric Graphs

@inproceedings{Daz1999LinearOO,
title={Linear Orderings of Random Geometric Graphs},
author={Josep D{\'i}az and Mathew D. Penrose and Jordi Petit and Maria J. Serna},
booktitle={WG},
year={1999}
}
• Published in WG 17 June 1999
• Mathematics, Computer Science
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…
9 Citations

### Layout Problems on Lattice Graphs

• Mathematics
COCOON
• 1999
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.

### A survey of graph layout problems

• Computer Science
CSUR
• 2002
A complete view of the current state of the art with respect to layout problems from an algorithmic point of view is presented.

### Building (1 - epsilon) Dominating Sets Partition as Backbones in Wireless Sensor Networks Using Distributed Graph Coloring

• Computer Science, Mathematics
DCOSS
• 2010
This paper demonstrates that by combining a local vertex ordering with the greedy color selection strategy, one can in practice, minimize the number of colors used to color an RGG within a very narrow window of the chromatic number and concurrently also obtain a domatic partition size within a competitive factor of the domatic number.

### Algorithms for the linear coloring arrangement problem

This project undertakes the task of developing efficient algorithms for solving or approximating the Minimum linear colouring arrangement problem for graphs (minlca), a variation of the Minimum

### Modeling interactome: scale-free or geometric?

• Computer Science
Bioinform.
• 2004
It is shown that the structure of PPI networks is better modeled by a geometric random graph than by a scale-free model, and a random geometric model provides a much more accurate model of the PPI data.

### Leading Edge Learning in Network Science

This article focuses on teaching and learning network science—the emerging science of studying complex networks by modeling its actors as nodes and their relationship as edges—through discovery-based

### Convergence Theorems for Some Layout Measures on Random Lattice and Random Geometric Graphs

• Mathematics, Computer Science
Combinatorics, Probability and Computing
• 2000
The 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.

## References

SHOWING 1-10 OF 31 REFERENCES

### Layout Problems on Lattice Graphs

• Mathematics
COCOON
• 1999
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.

### Approximating layout problems on random sparse graphs

• Mathematics, Computer Science
• 1998
We show that, with high probability, several layout problems are approximable within a constant for random graphs drawn from the standard Gnp model with p=c/n for some constant c. Our results

### Optimal numberings of an N N array

• Mathematics
• 1986
Given a numbering of the vertices of a graph, one can define the edgesum [6] as the sum of differences between adjacent vertices. The problem of finding numberings which are optimal in the sense of

### The bisection width of grid graphs

• Computer Science, Mathematics
SODA '90
• 1990
A polynomial algorithm for computing the minimum number of edges the authors need to delete in order to divide a given solid grid graph into two parts containing an equal number of nodes is presented.

### On Path-Tough Graphs

• Mathematics
SIAM J. Discret. Math.
• 1994
The authors prove that every graph of order n and minimum degree at least $$3/(6+\sqrt{3})$n$ is Hamiltonian if and only if it is path-tough.

### The longest edge of the random minimal spanning tree

For n points placed uniformly at random on the unit square, suppose Mn (respectively,Mn) denotes the longest edge-length of the nearest neighbor graph (respectively, the minimal spanning tree) on

### The Vertex Separation and Search Number of a Graph

• Mathematics, Computer Science
Inf. Comput.
• 1994
Algorithms that, for any tree T, compute vs ( T ) in linear time and compute an optimal layout with respect to vertex separation in time O ( n log n) are given.

### Eigenvalues and graph bisection: An average-case analysis

• R. Boppana
• Computer Science, Mathematics
28th Annual Symposium on Foundations of Computer Science (sfcs 1987)
• 1987
This paper presents an algorithm that will, for almost all graphs in a certain class, output the minimum-size bisection and will yield a proof that the bisection is optimal.