Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2connected graph. Cheriyan and Maheshwari gave an O(|V | 2 ) algorithm for finding three independent spanning trees in a 3-connected graph. In this paper we present an O(|V | 3 ) algorithm for finding four independent spanning trees in a 4-connected graph. We make use of chain decompositions of 4-connected graphs.

In 1978 Woodall [6] conjectured the following: in a planar digraph the size of a shortest cycle is equal to the maximum cardinality of a collection of disjoint tranversals of cycles. We prove that this conjecture is true when the digraph is series-parallel. In fact, we prove a stronger weighted version that gives the latter result as a corollary.

BackgroundDuring evolution, global mutations may alter the order and the orientation of the genes in a genome. Such mutations are referred to as rearrangement events, or simply operations. In unichromosomal genomes, the most common operations are reversals, which are responsible for reversing the order and orientation of a sequence of genes, and transpositions, which are responsible for switching the location of two contiguous portions of a genome. The problem of computing the minimum sequence of operations that transforms one genome into another – which is equivalent to the problem of sorting a permutation into the identity permutation – is a well-studied problem that finds application in comparative genomics. There are a number of works concerning this problem in the literature, but they generally do not take into account the length of the operations (i.e. the number of genes affected by the operations). Since it has been observed that short operations are prevalent in the evolution of some species, algorithms that efficiently solve this problem in the special case of short operations are of interest.ResultsIn this paper, we investigate the problem of sorting a signed permutation by short operations. More precisely, we study four flavors of this problem: (i) the problem of sorting a signed permutation by reversals of length at most 2; (ii) the problem of sorting a signed permutation by reversals of length at most 3; (iii) the problem of sorting a signed permutation by reversals and transpositions of length at most 2; and (iv) the problem of sorting a signed permutation by reversals and transpositions of length at most 3. We present polynomial-time solutions for problems (i) and (iii), a 5-approximation for problem (ii), and a 3-approximation for problem (iv). Moreover, we show that the expected approximation ratio of the 5-approximation algorithm is not greater than 3 for random signed permutations with more than 12 elements. Finally, we present experimental results that show that the approximation ratios of the approximation algorithms cannot be smaller than 3. In particular, this means that the approximation ratio of the 3-approximation algorithm is tight.

In this paper we give a decomposition of a 4-connected graph G into nonseparating chains, which is similar to an ear decomposition of a 2-connected graph. We also give an O(|V (G)| 2 |E(G)|) algorithm that constructs such a decomposition. In applications, the asymptotic performance can often be improved to O(|V (G)| 3). This decomposition will be used to find four independent spanning trees in a 4-connected graph.

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