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On the complexity of optimal drawings of graphs

  • Franz J. Brandenburg
Graph Embedding
Part of the Lecture Notes in Computer Science book series (LNCS, volume 411)

Abstract

We consider the problem of producing aesthetically nice drawings of graphs from the complexity point of view. The following questions are immediate:
  1. (1)

    How to formalize in algorithmic terms that a drawing is nice?

     
  2. (2)

    What are the computational costs for nice drawings?

     
  3. (3)

    Are there tools to beat the NP-completeness?

     

For (1) we propose grid embeddings of graphs and measure ”nice” by algorithmic cost measures of the embeddings, e.g., area, expansion, edge length, etc. For (2) we prove that optimal embeddings with fixed costs are NP-complete, even for binary trees. This sharpens previous NP-completeness results of optimal embeddings from connected graphs to binary trees and extends them to other cost measures. For (3) we introduce placement graph grammars. These are special graph grammars enriched by a placement component. The placement component contains partial information on the relative positions of the vertices, which is a reduction of the placement information contained in any concrete grid drawing. Every derivation of a graph in the base graph grammar has an associated placement component. By an extension of the parsing process we can compute a placement of the vertices of each generated graph, which is consistent with the associated placement component, and is area minimal. For connected graphs of bounded degree this can be done in polynomial time.

Keywords

graph layout embeddings NP-completeness graph grammars placement graph grammars 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Franz J. Brandenburg
    • 1
  1. 1.Lehrstuhl für InformatikUniversität PassauPassauF.R. Germany

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