Enumerating topological (nk)(n_k)-configurations

An $(n_k)$-configuration is a set of $n$ points and $n$ lines in the projective plane such that their point-line incidence graph is $k$-regular. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. We provide an algorithm for generating, for given $n$ and $k$, all topological $(n_k)$-configurations up to combinatorial isomorphism, without enumerating first all combinatorial $(n_k)$-configurations. We apply this algorithm to confirm efficiently a former result on topological $(18_4)$-configurations, from which we obtain a new geometric $(18_4)$-configuration. Preliminary results on $(19_4)$-configurations are also briefly reported.Comment: 18 pages, 11 figure

Face pairing graphs and 3-manifold enumeration

The face pairing graph of a 3-manifold triangulation is a 4-valent graph denoting which tetrahedron faces are identified with which others. We present a series of properties that must be satisfied by the face pairing graph of a closed minimal P^2-irreducible triangulation. In addition we present constraints upon the combinatorial structure of such a triangulation that can be deduced from its face pairing graph. These results are then applied to the enumeration of closed minimal P^2-irreducible 3-manifold triangulations, leading to a significant improvement in the performance of the enumeration algorithm. Results are offered for both orientable and non-orientable triangulations.Comment: 30 pages, 57 figures; v2: clarified some passages and generalised the final theorem to the non-orientable case; v3: fixed a flaw in the proof of the conical face lemm

CLEVER: Clique-Enumerating Variant Finder

Next-generation sequencing techniques have facilitated a large scale analysis of human genetic variation. Despite the advances in sequencing speeds, the computational discovery of structural variants is not yet standard. It is likely that many variants have remained undiscovered in most sequenced individuals. Here we present a novel internal segment size based approach, which organizes all, including also concordant reads into a read alignment graph where max-cliques represent maximal contradiction-free groups of alignments. A specifically engineered algorithm then enumerates all max-cliques and statistically evaluates them for their potential to reflect insertions or deletions (indels). For the first time in the literature, we compare a large range of state-of-the-art approaches using simulated Illumina reads from a fully annotated genome and present various relevant performance statistics. We achieve superior performance rates in particular on indels of sizes 20--100, which have been exposed as a current major challenge in the SV discovery literature and where prior insert size based approaches have limitations. In that size range, we outperform even split read aligners. We achieve good results also on real data where we make a substantial amount of correct predictions as the only tool, which complement the predictions of split-read aligners. CLEVER is open source (GPL) and available from http://clever-sv.googlecode.com.Comment: 30 pages, 8 figure