## Finding Hamiltonian Cycles in Delaunay Triangulations Is NP-Complete (1994)

Venue: | IN PROC. 4TH CANAD. CONF. COMPUT. GEOM |

Citations: | 19 - 1 self |

### BibTeX

@INPROCEEDINGS{Dillencourt94findinghamiltonian,

author = {Michael B. Dillencourt},

title = {Finding Hamiltonian Cycles in Delaunay Triangulations Is NP-Complete},

booktitle = {IN PROC. 4TH CANAD. CONF. COMPUT. GEOM},

year = {1994},

pages = {223--228},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

It is shown that it is an NP-complete problem to determine whether a Delaunay triangulation or an inscribable polyhedron has a Hamiltonian cycle. It is also shown that there exist nondegenerate Delaunay triangulations and simplicial, inscribable polyhedra without 2-factors.

### Citations

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Computational Geometry: An Introduction
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(Show Context)
Citation Context ...ath) in G is a cycle (path) in the planar dual of G. A cutset in a plane graph G corresponds, in a natural fashion, to a dual cycle. The Delaunay triangulation is the dual of the Voronoi diagram; see =-=[2, 16, 26]-=- for details. In particular, a Delaunay triangulation is nondegenerate if all interior faces are triangles. An inscribed polyhedron is a convex polyhedron all of whose vertices lie on a common sphere.... |

1385 |
Graph theory with applications
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- 1976
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Citation Context ...ove by showing that there exist inscribable polyhedra (and Delaunay triangulations) that fail to have 2-factors (Section 4). 2 Preliminaries Except as noted, we use the graph-theoretic terminology of =-=[4]-=-. V (G) and E(G) denote, respectively, the set of vertices and edges of a graph G. A Hamiltonian cycle in a graph 1 A graph G is k-Hamiltonian if any graph obtained by removing k vertices from G is Ha... |

481 | Computational geometry
- Shamos
- 1978
(Show Context)
Citation Context ...triangulations have perfect matchings (1-factors) [14]. The question of whether Delaunay triangulations necessarily have Hamiltonian cycles was posed in [22], [24], and, in a closely related form, in =-=[31]-=-. Counterexamples satisfying progressively more restrictive conditions can be found in [19], [11], and [12]. These counterexamples suggest the computational question: what is the computational complex... |

399 | How to draw a graph
- Tutte
- 1963
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Citation Context ... 1 1 1 33 33 33 1 1 1 1 1 1 64 40 28 34 64 34 28 64 40 Figure 5: Edge weightings for the graph of Figure 4. The absence of a 2-factor follows from the following special case of Tutte's factor theorem =-=[35]-=-: Theorem 4.1 A graph G fails to have a 2-factor if and only if the vertices of G can be partitioned into three sets R, S, and T such that 2jT j ! c e (R) + 2jSj \Gamma X s2S d S[R (s); (1) where j \D... |

292 |
Introduction to Geometry
- Coxeter
- 1969
(Show Context)
Citation Context ...nd connecting all vertices incident on f to the new vertex. The following lemma, which is closely related to a result in [5], is an easy consequence of standard properties of stereographic projection =-=[9]-=-. Lemma 2.1 A plane graph G is Delaunay realizable, with face f as its unbounded face, if and only if the graph obtained from G by stellating f is inscribable. Our proof also makes use of the followin... |

188 |
Voronoi Diagrams-A Survey of a Fundamental Geometric Data Structure
- Aurenhammer
- 1991
(Show Context)
Citation Context ...ath) in G is a cycle (path) in the planar dual of G. A cutset in a plane graph G corresponds, in a natural fashion, to a dual cycle. The Delaunay triangulation is the dual of the Voronoi diagram; see =-=[2, 16, 26]-=- for details. In particular, a Delaunay triangulation is nondegenerate if all interior faces are triangles. An inscribed polyhedron is a convex polyhedron all of whose vertices lie on a common sphere.... |

174 | Geometric structures for three-dimensional shape representation
- Boissonnat
- 1984
(Show Context)
Citation Context ...aunay triangulation as a starting point, can be found in [27, 32]. Applications of Hamiltonian cycles in Delaunay triangulations to problems in pattern recognition and solid modeling are discussed in =-=[3, 22, 24, 25]. From a m-=-ore theoretical viewpoint, there appears to be a close connection between the structure of inscribable polyhedra and Hamiltonian cycles. Hamiltonicity is "almost" sufficient for inscribabili... |

136 |
Algorithms in Combinatorial Geometry, volume 10
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(Show Context)
Citation Context ...ath) in G is a cycle (path) in the planar dual of G. A cutset in a plane graph G corresponds, in a natural fashion, to a dual cycle. The Delaunay triangulation is the dual of the Voronoi diagram; see =-=[2, 16, 26]-=- for details. In particular, a Delaunay triangulation is nondegenerate if all interior faces are triangles. An inscribed polyhedron is a convex polyhedron all of whose vertices lie on a common sphere.... |

102 |
A theorem on planar graphs
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Citation Context ...ble in a certain degenerate sense [10, Page 303]. More recently, Dillencourt and Smith have shown that any 1-Hamiltonian planar graph is inscribable [15]. 1 Since 4-connected graphs are 1-Hamiltonian =-=[34, 36]-=- and 2-Hamiltonian [33], it follows that all 4-connected planar graphs and all graphs obtained by deleting a single vertex from a 4-connected planar graph are inscribable. Conversely, empirical eviden... |

69 |
Voronöı diagrams from convex hulls
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Citation Context ...of stellating the face f consists of adding a new vertex in the interior of f and connecting all vertices incident on f to the new vertex. The following lemma, which is closely related to a result in =-=[5]-=-, is an easy consequence of standard properties of stereographic projection [9]. Lemma 2.1 A plane graph G is Delaunay realizable, with face f as its unbounded face, if and only if the graph obtained ... |

64 |
The-Four Color Problem
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Citation Context ...al, inscribable graphs. Let G be an n-vertex, 2-connected bipartite trivalent plane graph with isolated same-color separators. Two-color the vertices of G red and blue. Let L be the medial graph of G =-=[23]-=-. That is, the vertices of L are the midpoints of the edges of G, and two vertices of L are joined by an edge if and only if the corresponding edges of G are consecutive edges on a common face of G. S... |

42 |
A characterization of ideal polyhedra in hyperbolic 3-space
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Citation Context ...ts unbounded face, if and only if the graph obtained from G by stellating f is inscribable. Our proof also makes use of the following numerical characterization of inscribable polyhedra, due to Rivin =-=[28]-=- (also see [18, 29, 30]). Theorem 2.2 A graph G is inscribable if and only if it is planar, 3-connected, and weights w can be assigned to its edges such that: (W1) For each edge e, 0 ! w(e) ! 1=2. (W2... |

42 |
A theorem on paths in planar graphs
- Thomassen
- 1983
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Citation Context ...ble in a certain degenerate sense [10, Page 303]. More recently, Dillencourt and Smith have shown that any 1-Hamiltonian planar graph is inscribable [15]. 1 Since 4-connected graphs are 1-Hamiltonian =-=[34, 36]-=- and 2-Hamiltonian [33], it follows that all 4-connected planar graphs and all graphs obtained by deleting a single vertex from a 4-connected planar graph are inscribable. Conversely, empirical eviden... |

31 |
Realizability of Delaunay triangulations
- Dillencourt
- 1990
(Show Context)
Citation Context ...an with high probability [17]. It has also been shown that inscribable polyhedra and Delaunay triangulations have certain Hamiltonian-like properties. For example, Delaunay triangulations are 1-tough =-=[14]-=-. 2 This implies, in particular, that all Delaunay triangulations have perfect matchings (1-factors) [14]. The question of whether Delaunay triangulations necessarily have Hamiltonian cycles was posed... |

28 |
Fast heuristics for large geometric traveling salesman problems
- Reinelt
- 1992
(Show Context)
Citation Context ... be expected to be a good approximation for the Euclidean Traveling Salesman Cycle (ETSC). Heuristics for approximating the ETSC, using the Delaunay triangulation as a starting point, can be found in =-=[27, 32]-=-. Applications of Hamiltonian cycles in Delaunay triangulations to problems in pattern recognition and solid modeling are discussed in [3, 22, 24, 25]. From a more theoretical viewpoint, there appears... |

26 | 4-connected projective-planar graphs are Hamiltonian
- Thomas, Yu
- 1994
(Show Context)
Citation Context ... sense [10, Page 303]. More recently, Dillencourt and Smith have shown that any 1-Hamiltonian planar graph is inscribable [15]. 1 Since 4-connected graphs are 1-Hamiltonian [34, 36] and 2-Hamiltonian =-=[33]-=-, it follows that all 4-connected planar graphs and all graphs obtained by deleting a single vertex from a 4-connected planar graph are inscribable. Conversely, empirical evidence suggests that Delaun... |

25 |
A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere
- Hodgson, Rivin, et al.
- 1992
(Show Context)
Citation Context ...ce, if and only if the graph obtained from G by stellating f is inscribable. Our proof also makes use of the following numerical characterization of inscribable polyhedra, due to Rivin [28] (also see =-=[18, 29, 30]-=-). Theorem 2.2 A graph G is inscribable if and only if it is planar, 3-connected, and weights w can be assigned to its edges such that: (W1) For each edge e, 0 ! w(e) ! 1=2. (W2) For each vertex v, th... |

22 | On trivalent graphs
- Biggs, Smith
- 1971
(Show Context)
Citation Context ...ce, if and only if the graph obtained from G by stellating f is inscribable. Our proof also makes use of the following numerical characterization of inscribable polyhedra, due to Rivin [28] (also see =-=[18, 29, 30]-=-). Theorem 2.2 A graph G is inscribable if and only if it is planar, 3-connected, and weights w can be assigned to its edges such that: (W1) For each edge e, 0 ! w(e) ! 1=2. (W2) For each vertex v, th... |

19 | Intrinsic geometry of convex ideal polyhedra in hyperbolic 3-space, Analysis, algebra, and computers in mathematical research (Lule˚a
- Rivin
- 1992
(Show Context)
Citation Context ...ce, if and only if the graph obtained from G by stellating f is inscribable. Our proof also makes use of the following numerical characterization of inscribable polyhedra, due to Rivin [28] (also see =-=[18, 29, 30]-=-). Theorem 2.2 A graph G is inscribable if and only if it is planar, 3-connected, and weights w can be assigned to its edges such that: (W1) For each edge e, 0 ! w(e) ! 1=2. (W2) For each vertex v, th... |

18 |
Connect-the-dots: a new heuristic
- O'Rourke, Booth, et al.
- 1987
(Show Context)
Citation Context ...aunay triangulation as a starting point, can be found in [27, 32]. Applications of Hamiltonian cycles in Delaunay triangulations to problems in pattern recognition and solid modeling are discussed in =-=[3, 22, 24, 25]. From a m-=-ore theoretical viewpoint, there appears to be a close connection between the structure of inscribable polyhedra and Hamiltonian cycles. Hamiltonicity is "almost" sufficient for inscribabili... |

16 | Graph-theoretical conditions for inscribability and Delaunayrealizability
- Dillencourt, Smith
- 1994
(Show Context)
Citation Context ...bserved that any Hamiltonian polyhedron is inscribable in a certain degenerate sense [10, Page 303]. More recently, Dillencourt and Smith have shown that any 1-Hamiltonian planar graph is inscribable =-=[15]-=-. 1 Since 4-connected graphs are 1-Hamiltonian [34, 36] and 2-Hamiltonian [33], it follows that all 4-connected planar graphs and all graphs obtained by deleting a single vertex from a 4-connected pla... |

15 |
A Non-Hamiltonian, Nondegenerate Delaunay Triangulation
- Dillencourt
- 1987
(Show Context)
Citation Context ...ations necessarily have Hamiltonian cycles was posed in [22], [24], and, in a closely related form, in [31]. Counterexamples satisfying progressively more restrictive conditions can be found in [19], =-=[11]-=-, and [12]. These counterexamples suggest the computational question: what is the computational complexity of finding Hamiltonian cycles in Delaunay triangulations? There have been some partial result... |

12 |
NP-Completeness of the Hamiltonian Cycle Problem for Bipartite Graphs
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- 1980
(Show Context)
Citation Context ... NP, so it is only necessary to show NP-hardness. The reduction is from the recognition problem for Hamiltonian 2-connected bipartite trivalent planar graphs (H2BTP), which was shown to be NP-hard in =-=[1]-=-. 2 Our method extends the construction used by Chv'atal to show that the recognition problem for Hamiltonian maximal planar graphs is NP-hard [6, page 427]. Our construction is more delicate, because... |

12 | Hamiltonian cycles - Chvátal - 1985 |

7 |
Hamiltonian cycles in planar triangulations with no separating triangles
- Dillencourt
- 1990
(Show Context)
Citation Context ...ggest the computational question: what is the computational complexity of finding Hamiltonian cycles in Delaunay triangulations? There have been some partial results aimed at addressing this question =-=[7, 8, 10, 13, 20]-=-. In the present paper, we settle the computational question by showing that it is an NP-complete problem to determine whether there is a Hamiltonian cycle in a simplicial inscribable graph (Theorem 3... |

7 |
Traveling salesman cycles are not always subgraphs of Voronoi duals
- Kantabutra
- 1983
(Show Context)
Citation Context ...iangulations necessarily have Hamiltonian cycles was posed in [22], [24], and, in a closely related form, in [31]. Counterexamples satisfying progressively more restrictive conditions can be found in =-=[19]-=-, [11], and [12]. These counterexamples suggest the computational question: what is the computational complexity of finding Hamiltonian cycles in Delaunay triangulations? There have been some partial ... |

6 |
Hamiltonian cycles in Delaunay complexes
- Crapo, Laumond
- 1988
(Show Context)
Citation Context ...ggest the computational question: what is the computational complexity of finding Hamiltonian cycles in Delaunay triangulations? There have been some partial results aimed at addressing this question =-=[7, 8, 10, 13, 20]-=-. In the present paper, we settle the computational question by showing that it is an NP-complete problem to determine whether there is a Hamiltonian cycle in a simplicial inscribable graph (Theorem 3... |

5 |
Finding Hamiltonian cycles in certain planar graphs
- Cimikowski
- 1990
(Show Context)
Citation Context ...ggest the computational question: what is the computational complexity of finding Hamiltonian cycles in Delaunay triangulations? There have been some partial results aimed at addressing this question =-=[7, 8, 10, 13, 20]-=-. In the present paper, we settle the computational question by showing that it is an NP-complete problem to determine whether there is a Hamiltonian cycle in a simplicial inscribable graph (Theorem 3... |

4 |
Etude du caractère hamiltonien de delaunays aléatoires. Travail de semestre
- Genoud
- 1990
(Show Context)
Citation Context ...y deleting a single vertex from a 4-connected planar graph are inscribable. Conversely, empirical evidence suggests that Delaunay triangulations of moderate size are Hamiltonian with high probability =-=[17]-=-. It has also been shown that inscribable polyhedra and Delaunay triangulations have certain Hamiltonian-like properties. For example, Delaunay triangulations are 1-tough [14]. 2 This implies, in part... |

4 | Graphtool: A tool for interactive design and manipulation of graphs and graph algorithms - Leung, Dillencourt, et al. - 1994 |

3 |
Some problems in computational geometry
- Mathieu
- 1987
(Show Context)
Citation Context ...aunay triangulation as a starting point, can be found in [27, 32]. Applications of Hamiltonian cycles in Delaunay triangulations to problems in pattern recognition and solid modeling are discussed in =-=[3, 22, 24, 25]. From a m-=-ore theoretical viewpoint, there appears to be a close connection between the structure of inscribable polyhedra and Hamiltonian cycles. Hamiltonicity is "almost" sufficient for inscribabili... |

3 |
Computational geometry column 2
- O’Rourke
- 1987
(Show Context)
Citation Context |

2 |
An upper bound on the shortness exponent of inscribable graphs
- Dillencourt
- 1989
(Show Context)
Citation Context ...essarily have Hamiltonian cycles was posed in [22], [24], and, in a closely related form, in [31]. Counterexamples satisfying progressively more restrictive conditions can be found in [19], [11], and =-=[12]-=-. These counterexamples suggest the computational question: what is the computational complexity of finding Hamiltonian cycles in Delaunay triangulations? There have been some partial results aimed at... |

1 |
On certain Hamiltonian inner triangulations
- Cimikowski
- 1993
(Show Context)
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1 |
Connectivity of planar triangulations
- Laumond
- 1990
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1 |
Euclidean traveling salesman problems and Voronoi diagrams
- Jr
- 1992
(Show Context)
Citation Context ... be expected to be a good approximation for the Euclidean Traveling Salesman Cycle (ETSC). Heuristics for approximating the ETSC, using the Delaunay triangulation as a starting point, can be found in =-=[27, 32]-=-. Applications of Hamiltonian cycles in Delaunay triangulations to problems in pattern recognition and solid modeling are discussed in [3, 22, 24, 25]. From a more theoretical viewpoint, there appears... |