## Unit Disk Graph Recognition is NP-Hard (1993)

Venue: | Computational Geometry. Theory and Applications |

Citations: | 79 - 1 self |

### BibTeX

@ARTICLE{Breu93unitdisk,

author = {Heinz Breu and David G. Kirkpatrick},

title = {Unit Disk Graph Recognition is NP-Hard},

journal = {Computational Geometry. Theory and Applications},

year = {1993},

volume = {9}

}

### Years of Citing Articles

### OpenURL

### Abstract

Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have sphericity at most 3, is NP-hard. We show how this reduction can be extended to 3 dimensions, thereby showing that unit sphere graph recognition, or determining if a graph has sphericity 3 or less, is also NP-hard. We conjecture that K-sphericity is NP-hard for all fixed K greater than 1. 1 Introduction A unit disk graph is the intersection graph of a set of unit diameter closed disks in the plane. That is, each vertex corresponds to a disk in the plane, and two vertices are adjacent in the graph if the corresponding disks intersect. The set of disks is said to realize the graph. Of course, the unit of distance is not critical, since the disks realize the same graph even if the coordina...

### Citations

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Computers and Intractability: A guide to the theory of NP-completeness
- Garey, Johnson
- 1978
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Citation Context .... QUESTION: Does G have sphericity K, that is, does it have a K-dimensional realization? This paper reduces SATISFIABILITY to 2-sphericity. SATISFIABILITY is a common basis for NP-completeness proofs =-=[GJ79]-=- and is the first problem to have been proved NP-complete [Coo71]. Let U = fu 1 ; u 2 ; : : : ; umg be a set of Boolean variables. A clause c = fl 1 ; l 2 ; : : : ; l k g is a set of literals, which m... |

778 | The complexity of theorem-proving procedures
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Citation Context ...dimensional realization? This paper reduces SATISFIABILITY to 2-sphericity. SATISFIABILITY is a common basis for NP-completeness proofs [GJ79] and is the first problem to have been proved NP-complete =-=[Coo71]-=-. Let U = fu 1 ; u 2 ; : : : ; umg be a set of Boolean variables. A clause c = fl 1 ; l 2 ; : : : ; l k g is a set of literals, which may be negated, e.g., u i , or unnegated, e.g., u i , variables. A... |

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Frequency assignment: Theory and applications
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Citation Context ... recognition problem for unit disk graphs: given a graph, determine if it has a realization. Unit disk graphs have been used to model several physical problems, for example radio frequency assignment =-=[Hal80]-=- and ship-to-ship communications (attributed to Marc Lipman by [Rob91]). They have also been used as test cases for heuristic algorithms designed for arbitrary graphs [JAMS91]. More applications are d... |

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Unit disk graphs
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Citation Context ...-ship communications (attributed to Marc Lipman by [Rob91]). They have also been used as test cases for heuristic algorithms designed for arbitrary graphs [JAMS91]. More applications are described by =-=[CCJ90]-=- and [MHR92]. Submitted to Algorithmica y On leave from Hewlett-Packard Laboratories 1 2 Breu and Kirkpatrick There are several motivations for recognizing unit disk graphs. First, unit disk graphs ar... |

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Indifference Graphs
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Citation Context ... 2 Breu and Kirkpatrick There are several motivations for recognizing unit disk graphs. First, unit disk graphs are a natural generalization of unit interval graphs, also known as indifference graphs =-=[Rob68]-=-. These graphs are interval graphs that do not contain K 1;3 as an induced subgraph [Rob68]. They can therefore be recognized in polynomial time [FG65, BL76]. Secondly, unit disk graphs can admit more... |

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The combinatorial distance geometry approach to the calculation of molecular conformation
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- 1983
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Citation Context ...se problems require a realization of the graph. A third motivation is simply to determine if a graph is a unit disk graph, for example, to test the physical feasibility of a graph modelling molecules =-=[Hav82a]-=-. We show in this paper that the problem of determining if a graph has a realization is itself NP-hard. This answers an open question mentioned in [CCJ90] and [MHR92]. This result has a more general c... |

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Geometry based approximation for intersecting graphs
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- 1992
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Citation Context ...ications (attributed to Marc Lipman by [Rob91]). They have also been used as test cases for heuristic algorithms designed for arbitrary graphs [JAMS91]. More applications are described by [CCJ90] and =-=[MHR92]-=-. Submitted to Algorithmica y On leave from Hewlett-Packard Laboratories 1 2 Breu and Kirkpatrick There are several motivations for recognizing unit disk graphs. First, unit disk graphs are a natural ... |

1 |
Quo Vadis, Graph Theory
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- 1991
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Citation Context ... it has a realization. Unit disk graphs have been used to model several physical problems, for example radio frequency assignment [Hal80] and ship-to-ship communications (attributed to Marc Lipman by =-=[Rob91]-=-). They have also been used as test cases for heuristic algorithms designed for arbitrary graphs [JAMS91]. More applications are described by [CCJ90] and [MHR92]. Submitted to Algorithmica y On leave ... |