Level Planarity Testing in Linear Time

Jünger, Michael and Leipert, Sebastian and Mutzel, Petra (1998) Level Planarity Testing in Linear Time. [Preprint]

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Abstract

In a leveled directed acyclic graph G = (V,E) the vertex set V is partitioned into k <= |V| levels V1,V2,...,Vk such that for each edge (u,v) in E with u in Vi and v in Vj we have i < j. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level Vi, all v in Vi are drawn on the line li = {(x,k-i) | x in R}, the edges are drawn monotone with respect to the vertical direction, and no edges intersect except at their end vertices. If G has a single source, the test can be performed in O(|V|) time by an algorithm of Di Battista and Nardelli [Hierarchies and planarity theory. IEEE Trans. Systems Man Cybernet. 18 (1988), no. 6, 1035--1046] that uses the PQ-tree data structure introduced by Booth and Lueker [Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. System Sci. 13 (1976), no. 3, 335--379]. PQ-trees have also been proposed by Heath and Pemmaraju (1996a,1996b) to test level planarity of leveled directed acyclic graphs with several sources and sinks. It has been shown in Jünger, Leipert and Mutzel (1997) that this algorithm is not correct in the sense that it does not state correctly level planarity of every level planar graph. In this paper, we present a correct linear time level planarity testing algorithm that is based on two main new techniques that replace the incorrect crucial parts of the algorithm of Heath and Pemmaraju (1996a,1996b).

Item Type:Preprint
Classifications:M Methods > M.500 Layered
P Styles > P.480 Layered
G Algorithms and Complexity > G.770 Planarity Testing
P Styles > P.540 Planar
ID Code:18
Alternative Locations:http://e-archive.informatik.uni-koeln.de/321/

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References

Booth, K. and Lueker G. (1976) Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms. Journal of Computer and System Sciences, 13 (1) 335-379.

Chiba, N. and Nishizeki, T. and Abe, S. and Ozawa, T. (1985) A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30 54-76.

Di Battista, G. and Nardelli, E. (1988) Hierarchies and planarity theory. IEEE Transactions on systems, man, and cybernetics, 18 (6) 1035-1046.

Heath, L. S. and Pemmaraju, S. V. (1995) Recognizing leveled-planar dags in linear time. In: F. J. Brandenburg (ed.) Proc. Graph Drawing 95, volume 1027 of Lecture Notes in Computer Science, Springer Verlag, pp. 300-311.

Heath, L. S. and Pemmaraju, S. V. (1996) Stack and queue layouts of directed acyclic graphs: Part II. Technical report, Department of Computer Science, Virginia Polytechnic Institute & State University.

Jünger, M. and Leipert, S. and Mutzel, P. (1997) Pitfalls of using PQ-trees in Automatic Graph Drawing. In: G. Di Battista (ed.) Graph Drawing 97, volume 1353 of Lecture Notes in Computer Science, Springer Verlag, pp. 193-204.