A linear algorithm for resource four-partitioning four-connected planar graphs

Awal, Tanveer; Rahman, Md. Saidur

doi:10.1109/icelce.2010.5700745

Cited by 2 publications

(1 citation statement)

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“…This problem can be solved in quadratic time [33] and, if the graph is additionally planar, even in linear time [22]. As suggested in [37,1], the problem (as well as a related extension) can be solved with the aid of a non-separating ear decomposition. For planar graphs, it thus suffices with Observation 11 to compute just a canonical ordering, which simplifies previous algorithms considerably.…”

Section: Application 3: Planarity Testingmentioning

confidence: 99%

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

Schmidt¹

2016

SIAM J. Comput.

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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 in a 3-connected graph in time O(m), improving a result of Cheriyan and Maheshwari [J. Algorithms 9(4)], -improve the preprocessing time from O(n 2 ) to O(m) 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, -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), while the previous best running time is O(n 2 ) due to Suzuki et al. [IPSJ 31(5)].

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Section: Application 3: Planarity Testingmentioning

confidence: 99%