Elsevier

Journal of Algorithms

Volume 9, Issue 4, December 1988, Pages 507-537
Journal of Algorithms

Finding nonseparating induced cycles and independent spanning trees in 3-connected graphs

https://doi.org/10.1016/0196-6774(88)90015-6Get rights and content

Abstract

Given a vertex r of a 3-connected graph G, we show how to find three independent spanning trees of G rooted at r. Our proof is based on showing that every 3-connected graph has a nonseparating ear decomposition. This extends Whitney's characterisation that a graph is 2-connected iff it has an ear decomposition. We also show that a nonseparating ear decomposition can be constructed in O(VE) time, and hence, three independent spanning trees can be found in O(VE) time. We construct a nonseparating ear decomposition by solving the following problem at most V times. Given an edge tr and a vertex u of a 3-connected graph G, find a nonseparating induced cycle of G through tr and avoiding u. W. T. Tutte (Proc. London Math. Soc. 13 (1963), 743–767) first showed that such a cycle can always be found. We give a linear time algorithm for this.

References (19)

  • C Thomassen et al.

    Nonseparating induced cycles in graphs

    J. Combin. Theory Ser. B

    (1981)
  • W.T Tutte

    A theory of 3-connected graphs

    Nederl. Akad. Wetensch. Indag. Math.

    (1961)
  • B Bollobas

    Extremal Graph Theory

    (1978)
  • B Bollobas

    Graph Theory

    (1979)
  • D Dolev et al.

    A new look at fault tolerant network routing

  • S Even

    Graph Algorithms

    (1979)
  • F Harary

    Graph Theory

    (1972)
  • J.E Hopcroft et al.

    Dividing a graph into triconnected components

    SIAM J. Comput.

    (1973)
  • J.E Hopcroft et al.

    Efficient planarity testing

    J. Assoc. Comput. Mach.

    (1974)
There are more references available in the full text version of this article.

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    Citation Excerpt :

    Then there is an open ear decomposition that naturally depends on T (see e.g. [12]), and most of the algorithms above compute only st-numberings that are consistent with this special open ear decomposition. However, several applications (e.g. the ones in [2,5,13,11]) need more general st-numberings that are consistent with an arbitrary given open ear decomposition D. The following is a very simple and probably folklore approach to obtain these st-numberings by falling back on st-orientations (see e.g. [9, Section 3.1] and [13, Application 1]): Compute an open ear decomposition D, orient the first cycle from s to t, and orient every following ear such that acyclicity is preserved; since the reachability relation is always a poset, an appropriate orientation for the next ear is guaranteed to exist.

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