Mondshein Sequences (a.k.a. (2,1)-Orders)

19 Aug 2016  ·  Schmidt Jens M. ·

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. Surprisingly, this fundamental link between canonical orderings and non-separating ear decomposition has not been established before. Currently, the fastest known algorithm for computing a Mondshein 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 subquadratic time. After putting Mondshein's work into context, we present an algorithm that computes a Mondshein sequence in optimal time and space O(m). This improves the previous best running time by a factor of n. We illustrate the impact of this result by deducing linear-time algorithms for five other problems, for four out of which the previous best running times have been quadratic. In particular, we show how to - compute three independent spanning trees of a 3-connected graph in time O(m), - improve the preprocessing time from O(n^2) to O(m) for a data structure reporting 3 internally disjoint paths between any given vertex pair, - derive a very simple O(n)-time planarity test once a Mondshein sequence has been computed, - compute a nested family of contractible subgraphs of 3-connected graphs in time O(m), - compute a 3-partition in time O(m).

PDF Abstract
No code implementations yet. Submit your code now

Categories


Data Structures and Algorithms Discrete Mathematics Combinatorics

Datasets


  Add Datasets introduced or used in this paper