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Canonical orderings [STOC'88, FOCS'92] have been used as a key tool in graphdrawing, graph encoding and visibility representations for the last decades. Westudy a far-reaching generalization of canonical orderings to non-planar graphsthat 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 graphwhile the remaining vertices from i+1 to n induce a connected graph.Mondshein's sequence generalizes canonical orderings and became later andindependently known under the name non-separating ear decomposition.Surprisingly, this fundamental link between canonical orderings andnon-separating ear decomposition has not been established before. Currently,the fastest known algorithm for computing a Mondshein sequence achieves arunning time of O(nm); the main open problem in Mondshein's and follow-up workis to improve this running time to subquadratic time.  After putting Mondshein's work into context, we present an algorithm thatcomputes a Mondshein sequence in optimal time and space O(m). This improves theprevious best running time by a factor of n. We illustrate the impact of thisresult by deducing linear-time algorithms for five other problems, for four outof which the previous best running times have been quadratic. In particular, weshow how to - compute three independent spanning trees of a 3-connected graphin time O(m), - improve the preprocessing time from O(n^2) to O(m) for a datastructure reporting 3 internally disjoint paths between any given vertex pair,- derive a very simple O(n)-time planarity test once a Mondshein sequence hasbeen computed, - compute a nested family of contractible subgraphs of3-connected graphs in time O(m), - compute a 3-partition in time O(m).

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