Alexandru I. Tomescu

Unicyclic and bicyclic graphs having minimum degree distance

In this paper characterizations of connected unicyclic and bicyclic graphs in terms of the degree sequence, as well as the graphs in these classes minimal with respect to the degree distance are given.

rNA: a fast and accurate short reads numerical aligner

SUMMARYnThe advent of high-throughput sequencers (HTS) introduced the need of new tools in order to analyse the large amount of data that those machines are able to produce. The mandatory first step for a wide range of analyses is the alignment of the sequences against a reference genome. We present a major update to our rNA (randomized Numerical Aligner) tool. The main feature of rNA is the fact that it achieves an accuracy greater than the majority of other tools in a feasible amount of time. rNA executables and source codes are freely downloadable at http://iga-rna.sourceforge.net/.nnnCONTACTnvezzi@appliedgenomics.org; delfabbro@appliedgenomics.orgnnnSUPPLEMENTARY INFORMATIONnSupplementary data are available at Bioinformatics online.

A novel min-cost flow method for estimating transcript expression with RNA-Seq

BackgroundThrough transcription and alternative splicing, a gene can be transcribed into different RNA sequences (isoforms), depending on the individual, on the tissue the cell is in, or in response to some stimuli. Recent RNA-Seq technology allows for new high-throughput ways for isoform identification and quantification based on short reads, and various methods have been put forward for this non-trivial problem.ResultsIn this paper we propose a novel radically different method based on minimum-cost network flows. This has a two-fold advantage: on the one hand, it translates the problem as an established one in the field of network flows, which can be solved in polynomial time, with different existing solvers; on the other hand, it is general enough to encompass many of the previous proposals under the least sum of squares model. Our method works as follows: in order to find the transcripts which best explain, under a given fitness model, a splicing graph resulting from an RNA-Seq experiment, we find a min-cost flow in an offset flow network, under an equivalent cost model. Under very weak assumptions on the fitness model, the optimal flow can be computed in polynomial time. Parsimoniously splitting the flow back into few path transcripts can be done with any of the heuristics and approximations available from the theory of network flows. In the present implementation, we choose the simple strategy of repeatedly removing the heaviest path.ConclusionsWe proposed a new very general method based on network flows for a multiassembly problem arising from isoform identification and quantification with RNA-Seq. Experimental results on prediction accuracy show that our method is very competitive with popular tools such as Cufflinks and IsoLasso. Our tool, called Traph (Transcrips in gRAPHs), is available at: http://www.cs.helsinki.fi/gsa/traph/.

Set graphs. I. Hereditarily finite sets and extensional acyclic orientations

A graph G is said to be a set graph if it admits an acyclic orientation which is also extensional, in the sense that the out-neighborhoods of its vertices are pairwise distinct. Equivalently, a set graph is the underlying graph of the digraph representation of a hereditarily finite set. In this paper, we initiate the study of set graphs. On the one hand, we identify several necessary conditions that every set graph must satisfy. On the other hand, we show that set graphs form a rich class of graphs containing all connected claw-free graphs and all graphs with a Hamiltonian path. In the case of claw-free graphs, we provide a polynomial-time algorithm for finding an extensional acyclic orientation. Inspired by manipulations of hereditarily finite sets, we give simple proofs of two well-known results about claw-free graphs. We give a complete characterization of unicyclic set graphs, and point out two NP-complete problems closely related to the problem of recognizing set graphs. Finally, we argue that these three problems are solvable in linear time on graphs of bounded treewidth.

Set Graphs. III. Proof Pearl: Claw-Free Graphs Mirrored into Transitive Hereditarily Finite Sets

We report on the formalization of two classical results about claw-free graphs, which have been verified correct by Jacob T. Schwartz’s proof-checker Referee. We have proved formally that every connected claw-free graph admits (1) a near-perfect matching, (2) Hamiltonian cycles in its square. To take advantage of the set-theoretic foundation of Referee, we exploited set equivalents of the graph-theoretic notions involved in our experiment: edge, source, square, etc. To ease some proofs, we have often resorted to weak counterparts of well-established notions such as cycle, claw-freeness, longest directed path, etc.

Counting extensional acyclic digraphs

Transitive sets with n elements were counted by Peddicord in 1962, by the use of Ackermann@?s numeric encoding of a (hereditarily finite) set. In this paper we give a combinatorial interpretation of this number by counting extensional acyclic digraphs. In a similar constructive manner, we also obtain the number of weakly extensional acyclic digraphs with a given number of labeled sinks and a given number of sources, or with a given number of vertices of maximum rank.

Indexes for Jumbled Pattern Matching in Strings, Trees and Graphs

We consider how to index strings, trees and graphs for jumbled pattern matching when we are asked to return a match if one exists. For example, we show how, given a tree containing two colours, we can build a quadratic-space index with which we can find a match in time proportional to the size of the match. We also show how we need only linear space if we are content with approximate matches.

A randomized Numerical Aligner (rNA)

With the advent of new sequencing technologies able to produce an enormous quantity of short genomic sequences, new tools able to search for them inside a genomic reference sequence have emerged. Because of chemical reading errors or of the variability between organisms, one is interested in finding not only exact occurrences, but also occurrences with up to k mismatches. The contribution of this paper is twofold. On the one hand, we present a generalization of the classical Rabin-Karp string matching algorithm to solve the k-mismatch problem, with average complexity O(n+m) (n text and m pattern lengths, respectively). On the other hand, we show how to employ this idea in conjunction with an index over the text, allowing to search a pattern, with up to k mismatches, in time proportional to its length. This novel tool-rNA (randomized Numerical Aligner)-is in general faster and more accurate than other available tools like SOAP2, BWA, and BOWTIE. rNA executables and source code are freely available at http://iga-rna.sourceforge.net/.

Ranking, unranking and random generation of extensional acyclic digraphs

Extensional acyclic digraphs are acyclic digraphs whose vertices have pairwise different sets of out-neighbors; they represent hereditarily finite sets, which stand at the basis of some computer languages. In this paper we give an O(n^3) algorithm for generating uniformly at random an extensional acyclic digraph on n vertices. This is done by first proposing a linear-time algorithm for encoding such digraphs by particular (n-1)-tuples of subsets of {0,...,n-2}. We then give a new counting recurrence for such tuples, which we exploit in ranking/unranking algorithms. These are also useful for indexing data structures by hereditarily finite sets.

A simpler proof for vertex-pancyclicity of squares of connected claw-free graphs

Abstract We give a simpler proof of the well-known result of Matthews and Sumner stating that squares of connected claw-free graphs are vertex-pancyclic. Contrary to the previous proof, our approach does not resort to Fleischner’s result stating that, when restricted to squares of graphs, vertex-pancyclicity and Hamiltonicity are equivalent. The same proof idea already yielded that connected claw-free graphs of even order have a perfect matching, which is another result of Sumner. We conclude by observing that this proof identifies a larger collection of graphs for which the two properties in question hold.