## Abstract

Let S ⊂ R

2 be a set of n point sites, where each

s ∈ S has an associated radius rs > 0. The disk graph

D(S) of S is the graph with vertex set S and an edge

between two sites s and t if and only if |st| ≤ rs + rt,

i.e., if the disks with centers s and t and radii rs and

rt, respectively, intersect. Disk graphs are useful to

model sensor networks.

We study the problems of finding triangles and of

computing the girth in disk graphs. These problems

are notoriously hard for general graphs, but better

solutions exist for special graph graph classes, such

as planar graphs. We obtain similar results for disk

graphs. In particular, we observe that the unweighted

girth of a disk graph can be computed in O(n log n)

worst-case time and that a shortest (Euclidean) triangle in a disk graph can be found in O(n log n) expected

time.

2 be a set of n point sites, where each

s ∈ S has an associated radius rs > 0. The disk graph

D(S) of S is the graph with vertex set S and an edge

between two sites s and t if and only if |st| ≤ rs + rt,

i.e., if the disks with centers s and t and radii rs and

rt, respectively, intersect. Disk graphs are useful to

model sensor networks.

We study the problems of finding triangles and of

computing the girth in disk graphs. These problems

are notoriously hard for general graphs, but better

solutions exist for special graph graph classes, such

as planar graphs. We obtain similar results for disk

graphs. In particular, we observe that the unweighted

girth of a disk graph can be computed in O(n log n)

worst-case time and that a shortest (Euclidean) triangle in a disk graph can be found in O(n log n) expected

time.

Original language | American English |
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Title of host publication | Proc. 33rd European Workshop on Computational Geometry, EuroCG 2017 |

Pages | 205-208 |

Number of pages | 4 |

State | Published - 2017 |

### Publication series

Name | Proc. 33rd European Workshop Comput. Geom.(EWCG), |
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