Zhi-Zhong Chen, Michelangelo Grigni, and Christos Papadimitriou. Planar map graphs. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC-98), pages 514--523, New York, 1998. ACM Press.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....suited for applications dealing with spatial regions which cannot partially overlap. Obvious examples of this kind of regions are regions that correspond to solid state physical objects, or geographic regions such as countries or administrative districts (this observation is made also in [34] and [11], where similar sets of topological relations are studied) For example, consider representing a hierarchy of geographical spatial information like topological constraints between regions of the same state, between a region and its state, between states of the same country, and between a state and ....

Z. Chen, M. Grigni, and C. Papadimitriou. Planar map graphs. In 30th Annual ACM Symp. on the Theory of Computing, 1998.

.... two vertices are adjacent In other words, is G isomorphic to the intersection graph of a set of simply connected regions in the plane This deceptively simple extension of propositional logic and its generalizations are often referred to in the literature as topological inference problems [CGP98a], CGP98b] CHK99] They have proved to be relevant in the area of geographic information systems [E93] EF91] and in graph drawing [DETT99] In spite of many e orts [K91a] K98] and false claims [SP92] ES93] no algorithm was found for their solution. It is known that these problems are at ....

Z.-Z. Chen, M. Grigni, and C. H. Papadimitriou, Planar map graphs, in: STOC '98, ACM, 1998, 514-523.

..... The problem of recognizing intersection graphs of planar curves (the socalled string graph problem ) is known to be NP hard [K91] but it is open whether this problem is decidable [KM91] In some very special cases, e.g. when S consists of segments, there are trivial recognition algorithms [CGP98], FMP95] But even in these cases we do not know much about the structure of intersection graphs. One of the most striking examples illustrating our ignorance in the subject is the following simple open Problem. Is it true that every planar graph is the intersection graph of a system of segments ....

Z.-Z. Chen, M. Grigni, and C. H. Papadimitriou, Planar map graphs, in: STOC '98, ACM, 1998, 514-523.

....with no three pairwise crossing edges and n vertices, the number of edges is in O(n) and calls such graphs quasi planar . For general k, see also [PSS94, PSS96] and [Val97, Val98] for recent work and further references. A. Liebers, Planarizing Graphs , JGAA, 5(1) 1 74 (2001) 11 Chen et al. CGP98] study intersection graphs of planar regions with disjoint interiors and call them planar map graphs . This generalizes planar graphs since planar graphs may be defined as the intersection graphs of planar regions with disjoint interiors such that no four regions meet at a point. Yet another way ....

Zhi-Zhong Chen, Michelangelo Grigni, and Christos H. Papadimitriou. Planar Map Graphs. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, STOC'98, pages 514--523, 1998.

....interesting for applications dealing with spatial regions which cannot overlap. Obvious examples for this kind of regions are geographic regions such as countries or administrative districts, but also regions which correspond to physical objects (this observation has also been made in [23] and [7] where similar sets of topological relations have been studied) The paper is organized as follows. Section 2 introduces RCC 8 and gives the necessary background; Sections 3 and 4 deal with the combination of topological and qualitative size constraints; Section 5 gives our results regarding the ....

Z. Chen, M. Grigni, and C. Papadimitriou. Planar map graphs. In 30th Annual ACM Symp. on the Theory of Computing, 1998.

.... 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]. y 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. z Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322. E mail: ....

....algorithm must somehow break symmetry to find a map for this graph. j (1) e (3) j e f g c b h i a a c i g b d f e h j d d g f b a c i h (2) Figure 1.3: A symmetric map graph, a map, and a witness. 3 1. 3 Summary of Results In the preliminary version of this paper [4] we gave an NP characterization of map graphs, and we also sketched a polynomial time recognition algorithm for 4 map graphs. In this paper we prove the first result, but for reasons of brevity we present a simpler variant of the second result. Specifically, we present a polynomial time ....

Z.-Z. Chen, M. Grigni, and C. Papadimitriou. Planar map graphs. Proc. 30th Ann. ACM Symp. Theory of Computing (STOC), 514--523, 1998.

....complicated cases we identify a way of recursing on a similar maximal clique, albeit in a smaller graph. The case analysis involved is very tedious (over a hundred cases must be examined) in Section 3 we include a top level summary without detailed proofs; for a draft of the complete proof see [1]. The objects studied in the case analysis are partial maps, that is, sets of planar regions corresponding to the part of the graph being examined, with space left for embedding the rest. We refine the maps by bringing in more regions until we reach a final map, one in which all unoccupied holes ....

....graph that is not 4 planar ) and hence polynomial time recognition does not follow from first principles. 3 Recognition of 4 Planar Graphs In this section we sketch the proof of our main result: Theorem 5 4 planar graphs can be recognized in polynomial time. For a draft of the full proof see [1]. 3.1 Preliminaries Let G be a graph. A map L is a finite set of planar regions that are disc homeomorphs with disjoint interiors. A map is a realization of G (or a map of G) if its regions are in one toone correspondence to the vertices of G, and in which two regions touch each other iff the ....

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Z. Chen, M. Grigni. C. H. Papadimitriou "Planar map graphs," manuscript, available at http://www.mathcs.emory.edu/¸mic/pmg/, 56 pp.

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Zhi-Zhong Chen, Michelangelo Grigni, and Christos Papadimitriou. Planar map graphs. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC-98), pages 514--523, New York, 1998. ACM Press.

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Zhi-Zhong Chen, Michelangelo Grigni, and Christos Papadimitriou. Planar map graphs. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC-98), pages 514--523, New York, 1998. ACM Press.

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