### Abstract

A caterpillar network (or graph) G is a tree with the property that removal of the leaf edges of G leaves one with a path. Here we focus on minimum weight spanning caterpillars where the vertices are points in the Euclidean plane and the costs of the path edges and the leaf edges are multiples of their corresponding Euclidean lengths. The flexibility in choosing the weight for path edges versus the weight for leaf edges gives some useful flexibility in modeling. In particular, one can accommodate problems motivated by communications theory such as the "last mile problem." Geometric and probabilistic inequalities are developed that lead to a limit theorem that is analogous to the well-known Beardwood, Halton, and Hammersley theorem for the length of the shortest tour through a random sample, but the minimal spanning caterpillars fall outside the scope of the theory of subadditive Euclidean functionals.

Original language | English (US) |
---|---|

Pages (from-to) | 992-1004 |

Number of pages | 13 |

Journal | Mathematics of Operations Research |

Volume | 40 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2015 |

Externally published | Yes |

### Fingerprint

### Keywords

- Beardwood Halton and Hammersley theorem
- Caterpillar graphs
- Euclidean networks
- Gutman graphs
- Minimal spanning trees
- Shortest paths
- Subadditive Euclidean functional
- Traveling salesman problem

### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science Applications
- Management Science and Operations Research

### Cite this

*Mathematics of Operations Research*,

*40*(4), 992-1004. https://doi.org/10.1287/moor.2014.0706

**Euclidean networks with a backbone and a limit theorem for minimum spanning caterpillars.** / Jevtic, Petar; Steele, J. Michael.

Research output: Contribution to journal › Article

*Mathematics of Operations Research*, vol. 40, no. 4, pp. 992-1004. https://doi.org/10.1287/moor.2014.0706

}

TY - JOUR

T1 - Euclidean networks with a backbone and a limit theorem for minimum spanning caterpillars

AU - Jevtic, Petar

AU - Steele, J. Michael

PY - 2015/1/1

Y1 - 2015/1/1

N2 - A caterpillar network (or graph) G is a tree with the property that removal of the leaf edges of G leaves one with a path. Here we focus on minimum weight spanning caterpillars where the vertices are points in the Euclidean plane and the costs of the path edges and the leaf edges are multiples of their corresponding Euclidean lengths. The flexibility in choosing the weight for path edges versus the weight for leaf edges gives some useful flexibility in modeling. In particular, one can accommodate problems motivated by communications theory such as the "last mile problem." Geometric and probabilistic inequalities are developed that lead to a limit theorem that is analogous to the well-known Beardwood, Halton, and Hammersley theorem for the length of the shortest tour through a random sample, but the minimal spanning caterpillars fall outside the scope of the theory of subadditive Euclidean functionals.

AB - A caterpillar network (or graph) G is a tree with the property that removal of the leaf edges of G leaves one with a path. Here we focus on minimum weight spanning caterpillars where the vertices are points in the Euclidean plane and the costs of the path edges and the leaf edges are multiples of their corresponding Euclidean lengths. The flexibility in choosing the weight for path edges versus the weight for leaf edges gives some useful flexibility in modeling. In particular, one can accommodate problems motivated by communications theory such as the "last mile problem." Geometric and probabilistic inequalities are developed that lead to a limit theorem that is analogous to the well-known Beardwood, Halton, and Hammersley theorem for the length of the shortest tour through a random sample, but the minimal spanning caterpillars fall outside the scope of the theory of subadditive Euclidean functionals.

KW - Beardwood Halton and Hammersley theorem

KW - Caterpillar graphs

KW - Euclidean networks

KW - Gutman graphs

KW - Minimal spanning trees

KW - Shortest paths

KW - Subadditive Euclidean functional

KW - Traveling salesman problem

UR - http://www.scopus.com/inward/record.url?scp=84947061945&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84947061945&partnerID=8YFLogxK

U2 - 10.1287/moor.2014.0706

DO - 10.1287/moor.2014.0706

M3 - Article

AN - SCOPUS:84947061945

VL - 40

SP - 992

EP - 1004

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

SN - 0364-765X

IS - 4

ER -