## NP-Completeness Results for Minimum Planar Spanners

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Citations: | 23 - 0 self |

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@MISC{Brandes_np-completenessresults,

author = {Ulrik Brandes and Dagmar Handke},

title = {NP-Completeness Results for Minimum Planar Spanners},

year = {}

}

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### Abstract

For any fixed parameter t _> 1, a t-spanner of a graph G is a spanning subgraph in which the distance between every pair of vertices is at most t times their distance in G. A minimum t-spanner is a t-spanner with minimum total edge weight or, in unweighted graphs, minimum number of edges. In this paper, we prove the AlP-hardness of finding minimum t-spanners for planar weighted graphs and digraphs if t _> 3, and for planar unweighted graphs and digraphs if t _> 5. We thus extend results on that problem to the interesting case where the instances are known to be planar. We also introduce the related problem of finding minimum planar t-spanners and establish its Alp-hardness for similar fixed values of t.

### Citations

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154 |
Graph spanners
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(Show Context)
Citation Context ...nterested in finding a sparse subnetwork that nevertheless guarantees constant delay factors. A survey of some results on the existence and efficient constructibility of (sparse) spanners is given in =-=[PS89]-=-. Further results and discussions concerning t--spanners and variants thereof can be found in [Soa92]. In most applications the sparseness of a spanner is crucial. The problem of finding t--spanners w... |

127 |
An optimal synchronizer for the hypercube
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- 1989
(Show Context)
Citation Context ...s. Then the simple path v 1 = 1; v 2 = 2; : : : ; v n = n forms a minimum spanning tree yielding a stretch factor of t = n \Gamma 1. The concept of spanners has been introduced by Peleg and Ullman in =-=[PU87]-=-, where they used spanners to synchronize asynchronous networks. One of many other applications for spanners are communication networks, where one is interested in finding a sparse subnetwork that nev... |

57 | Tree spanners
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- 1995
(Show Context)
Citation Context ...r example that similar results for the tree t--spanner problem in general unweighted, undirected graphs show that this problem is NP--complete for ts4, but polynomially solvable for t equal to 1 or 2 =-=[CC95]-=-. In particular, the complexity status is left open for the following interesting problems: ffl MPS t for unweighted, undirected graphs with t 2 f2; 3g ffl M(P)S t for unweighted, planar, undirected g... |

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33 |
NP-completeness of minimum spanner problems
- Cai
(Show Context)
Citation Context ...formatik, Postfach 5560/D188, 78434 Konstanz, Germany. fUlrik.Brandes,Dagmar.Handkeg@uni-konstanz.de http://www.informatik.uni-konstanz.de/~fbrandes,handkeg/ 1 2 PROBLEMS AND RESULTS 2 of t by Cai in =-=[Cai94]-=-. Therefore subsequent efforts have concentrated on finding spanners that are maybe not minimum, but sufficiently sparse (see for example [ADD + 93]). Several authors considered variants of t--spanner... |

24 |
Determining the thickness of a graph is NP-hard
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- 1983
(Show Context)
Citation Context ...te graph G = (V; E) where V = U [ C and E = ffx; cg : x or x occurs in cg is planar. Problem: Is there a satisfying truth assignment for C? The NP--completeness proof for this problem can be found in =-=[Man83]-=-. We use the planarity of the underlying graph of P3SAT to construct a planar graph in which we can easily determine the minimum spanner. 3.2 Forcing edges into a spanner To force an edge into a spann... |

9 | Pandu Rangan, Restrictions of minimum spanner problems, Inform Comput 136(2 - Venkatesan, Rotics, et al. - 1997 |

8 |
Graph spanners: A survey
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(Show Context)
Citation Context ...ey of some results on the existence and efficient constructibility of (sparse) spanners is given in [PS89]. Further results and discussions concerning t--spanners and variants thereof can be found in =-=[Soa92]-=-. In most applications the sparseness of a spanner is crucial. The problem of finding t--spanners with a minimum number of edges has been shown to be NP--hard for most values The authors acknowledge t... |

6 |
Grid spanners
- Liestman, Shermer
- 1993
(Show Context)
Citation Context ...ity status of the corresponding decision problem. Liestman and Shermer introduced the notion of additive spanners, which employ an additive instead of multiplicative stretch function on the distances =-=[LS93]-=-. Here we consider spanners in planar graphs (either weighted or unweighted, directed or undirected), i.e. we restrict the set of input instances. We thereby (partially) settle a question raised in [C... |

2 |
Distance matrix of a graph and its
- Hakimi, Yau
- 1964
(Show Context)
Citation Context ...eal value W . Problem: Does G contain a t--spanner with total edge weight at most W ? Obviously, for an unweighted graph, the only 1--spanner is the graph itself. For a weighted graph, Hakimi and Yau =-=[HY64]-=- proved that there is a unique 1--spanner with a minimal number of edges. From [CC95] we know that this must also be the unique minimum 1-- spanner, and that it can be determined in polynomial time. T... |

1 | Determining the thickness of graphs is - Mansfield - 1983 |

1 | Grid spanners. Networks - Liestman, Shermer - 1993 |

1 | Graph spanners. J. of Graph Theory - Peleg, Schkffer - 1989 |