# 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} }

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…

## 10 Citations

### Layout Problems on Lattice Graphs

- MathematicsCOCOON
- 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 ScienceCSUR
- 2002

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

### Approximating Layout Problems on Random Geometric Graphs

- Mathematics, Computer ScienceJ. Algorithms
- 2001

This paper proves that some of the layout problems on a family of random geometric graphs remain NP-complete even for geometric graphs, and presents two heuristics that, almost surely, turn out to be constant approximation algorithms for their layout problems.

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

- Mathematics, Computer ScienceCombinatorics, 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.

### Vertex order with optimal number of adjacent predecessors

- Mathematics, Computer ScienceDiscret. Math. Theor. Comput. Sci.
- 2020

The complexity of the selection of a graph discretization order with a stepwise linear cost function is studied and it is found that the problem is NP-complete in general for all values of K and U such that U ≥ K + 1 and U ≥ 2.

### Optimal Cheeger cuts and bisections of random geometric graphs

- MathematicsThe Annals of Applied Probability
- 2020

Let $d \geq 2$. The Cheeger constant of a graph is the minimum surface-to-volume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and…

### PR ] 22 M ay 2 01 8 Optimal Cheeger cuts and bisections of random geometric graphs

- Mathematics
- 2018

Let d ≥ 2. The Cheeger constant of a graph is the minimum surface-tovolume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and volume…

### Algebraic topology of random simplicial complexes and applications to sensor networks

- Mathematics
- 2012

Stochastic anlysis is used to provide bounds for the overload probability of diﬀerent systems thanks to concentration inequalities and analytical results for quantites such as the distribution of the number of connected components and the probability of complete coverage are obtained.

### Random channel assignment in the plane

- MathematicsRandom Struct. Algorithms
- 2003

This work considers the first n random points, and is interested in particular in the behavior of the ratio of the chromatic number to the clique number, and shows that, as n → ∞, in probability this ratio tends to 1 in the "sparse" case and tends to 2√3/π ˜ 1.103 in the 'dense' case.

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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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This paper proves that some of the layout problems on a family of random geometric graphs remain NP-complete even for geometric graphs, and presents two heuristics that, almost surely, turn out to be constant approximation algorithms for their layout problems.

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

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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.

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