## Map Graphs (1999)

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Citations: | 26 - 3 self |

### BibTeX

@MISC{Chen99mapgraphs,

author = {Zhi-Zhong Chen and Michelangelo Grigni and Christos H. Papadimitriou},

title = {Map Graphs},

year = {1999}

}

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### Abstract

We consider a modified notion of planarity, in which two nations of a map are considered adjacent when they share any point of their boundaries (not necessarily an edge, as planarity requires). Such adjacencies define a map graph. We give an NP characterization for such graphs, and an O(n³)-time recognition algorithm for a restricted version: given a graph, decide whether it is realized by adjacencies in a map without holes, in which at most four nations meet at any point.

### Citations

2373 | Maintaining knowledge about temporal intervals
- Allen
- 1983
(Show Context)
Citation Context ...problem [9]. No decision algorithm or finite axiomatization is known, although the problem becomes both finitely axiomatizable and polynomial-time decidable in any number of dimensions other than two =-=[1, 14]-=-, for some reasonable vocabularies of topological relations. In fact, the following special case has been open since the 1960’s [8]: for every pair of regions, we are only told whether the regions int... |

231 | Efficient planarity testing
- Hopcroft, Karp
- 1974
(Show Context)
Citation Context ...ertices. Since H is simple and bipartite, by Euler’s formula it has at most 2(n + |P|) − 4 edges, which is less than 8n. ✷ In particular, a map graph has a witness which may be checked in linear time =-=[10]-=-, and so we have: Corollary 2.4 The recognition problem for map graphs is in NP. Let α(G) denote the arboricity of a graph G, the minimum number of forests whose union is G. The next result is useful ... |

78 | Topological Inference
- Grigni, Papadias, et al.
- 1995
(Show Context)
Citation Context ...ons possible? If so, we would like a model, a picture of four regions so related; if not, a proof of impossibility. This extension of propositional logic is known as the topological inference problem =-=[9]-=-. No decision algorithm or finite axiomatization is known, although the problem becomes both finitely axiomatizable and polynomial-time decidable in any number of dimensions other than two [1, 14], fo... |

72 |
Arboricity and subgraph listing algorithms
- Chiba, Nishizeki
- 1985
(Show Context)
Citation Context ... solution from the subproblem solutions, so we ignore the latter. By Corollary 2.5 (with k = 4) G has m = O(n) edges and arboricity α(G) = O(1), so we can list its O(n) maximal cliques in linear time =-=[7]-=-. From the listed MC4’s, we can precompute the sets E[a,b] for all unmarked edges {a,b}, again in linear time. We claim that testing the existence of a separating triangle takes O(n2 ) time. Since G h... |

59 |
Nash-Williams. Edge-disjoint spanning trees of finite graphs
- A
- 1961
(Show Context)
Citation Context ..., and consequently R has degree less than 7k in the map graph. Now we delete R, and prove our edge bound by induction on n. ⌉ where U ranges over all subsets of V (G) containing at least two vertices =-=[13]-=-, and each G[U] is again a k-map graph, the edge bound implies the arboricity bound. ✷ Since α(G) = maxU⌈ |E(G[U])| |U|−1 5For the simplicity of our figures, we prefer to draw maps rather than planar... |

46 | A Canonical Model of the Region Connection Calculus
- Renz
- 2002
(Show Context)
Citation Context ...problem [9]. No decision algorithm or finite axiomatization is known, although the problem becomes both finitely axiomatizable and polynomial-time decidable in any number of dimensions other than two =-=[1, 14]-=-, for some reasonable vocabularies of topological relations. In fact, the following special case has been open since the 1960’s [8]: for every pair of regions, we are only told whether the regions int... |

34 | Intersection graphs of curves in the plane - Ehrlich, Even, et al. - 1976 |

29 |
graphs II. Recognizing string graphs in NP-hard
- Kratochvil, String
(Show Context)
Citation Context ...orithm for the intersection graphs of planar curves. It is open whether this problem is decidable; it is known that there are infinitely many forbidden subgraphs, that recognition is at least NP-hard =-=[11]-=-, and that there are string graphs that require exponentially many string intersections for their realization [12]. ∗ A preliminary version appeared with the title “Planar Map Graphs” [4]. † Departmen... |

24 |
String graphs requiring exponential representations
- Kratochvíl, Matou˘sek
- 1991
(Show Context)
Citation Context ...t there are infinitely many forbidden subgraphs, that recognition is at least NP-hard [11], and that there are string graphs that require exponentially many string intersections for their realization =-=[12]-=-. ∗ A preliminary version appeared with the title “Planar Map Graphs” [4]. † Department of Mathematical Sciences, Tokyo Denki University, Hatoyama, Saitama 350-0394, JAPAN. Part of work done while vis... |

17 |
A new proof of the 6-color theorem
- Borodin
- 1995
(Show Context)
Citation Context ...ive because of the famous Four Color Theorem. As Thorup observed [15], the answer is also positive for k = 4: the 4-map graphs are all 1-planar graphs, and 1-planar graphs are known to be 6-colorable =-=[2]-=-. However, the answer is unknown when k ≥ 5. Similarly, we are interested in tighter versions of the bounds in Section 2; in particular the first author [3] has improved the edge bound in Corollary 2.... |

16 |
Map graphs in polynomial time
- Thorup
- 1998
(Show Context)
Citation Context ...gument for the case with holes is correct, however it is a very long case analysis which would double the length of the present paper. We left general map graph recognition as an open problem; Thorup =-=[15]-=- has recently presented a polynomial-time algorithm for recognizing map graphs. Thorup’s result does not necessarily imply ours, since even if we are given a map realizing a map graph, it is not clear... |

13 | Planar map graphs
- Chen, Grigni, et al.
- 1998
(Show Context)
Citation Context ...ast NP-hard [11], and that there are string graphs that require exponentially many string intersections for their realization [12]. ∗ A preliminary version appeared with the title “Planar Map Graphs” =-=[4]-=-. † Department of Mathematical Sciences, Tokyo Denki University, Hatoyama, Saitama 350-0394, JAPAN. Part of work done while visiting University of California at Berkeley. E-mail: chen@r.dendai.ac.jp. ... |

12 |
Approximation algorithms for independent sets in map graphs
- Chen
- 2001
(Show Context)
Citation Context ...1-planar graphs are known to be 6-colorable [2]. However, the answer is unknown when k ≥ 5. Similarly, we are interested in tighter versions of the bounds in Section 2; in particular the first author =-=[3]-=- has improved the edge bound in Corollary 2.5 to kn − 2k. The recognition problem of map graphs is just a special topological inference problem, where each pair of regions either touch or are disjoint... |

6 | Nonplanar topological inference and political-map graphs - Chen, He, et al. - 1999 |

6 | E cient Planarity Testing - Hopcroft, Tarjan - 1974 |

3 | il. String graphs II: Recognizing string graphs is NP-hard - Kratochv - 1991 |

3 | String graphs requiring exponential representations - l, ek - 1991 |

1 | Hierarchical topological inference on planar disc maps - Chen, He - 1999 |