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Abstract

Canonical orderings [STOC’88, FOCS’92] have been used as a key tool in graph drawing, graph encoding and visibility representations for the last decades. We study a far-reaching generalization of canonical orderings to non-planar graphs that was published by Lee Mondshein in a PhD-thesis at M.I.T. as early as 1971.

Mondshein proposed to order the vertices of a graph in a sequence such that, for any i, the vertices from 1 to i induce essentially a 2-connected graph while the remaining vertices from i + 1 to n induce a connected graph. Mondshein’s sequence generalizes canonical orderings and became later and independently known under the name non-separating ear decomposition. Currently, the best known algorithm for computing this sequence achieves a running time of O(nm); the main open problem in Mondshein’s and follow-up work is to improve this running time to a subquadratic time.

In this paper, we present the first algorithm that computes a Mondshein sequence in time and space O(m), improving the previous best running time by a factor of n. In addition, we illustrate the impact of this result by deducing linear-time algorithms for several other problems, for which the previous best running times have been quadratic.

In particular, we show how to compute three independent spanning trees in a 3-connected graph in linear time, improving a result of Cheriyan and Maheshwari [J. Algorithms 9(4)]. Secondly, we improve the preprocessing time for the output-sensitive data structure by Di Battista, Tamassia and Vismara [Algorithmica 23(4)] that reports three internally disjoint paths between any given vertex pair from O(n 2) to O(m). Thirdly, we improve the computation of 3-partitioning of a 3-connected graph to linear time. Finally, we show how a very simple linear-time planarity test can be derived once a Mondshein sequence is computed.

Keywords

Span Tree Planar Graph Disjoint Path Outer Face Graph Drawing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Jens M. Schmidt
    • 1
  1. 1.TU IlmenauGermany

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