Finding Biclique Partitions of Co-Chordal Graphs

The biclique partition number $(\text{bp})$ of a graph $G$ is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the biclique partition number ($\text{bp}$) of a co-chordal (complementary graph of chordal) graph $G = (V, E)$ is less than the number of maximal cliques ($\text{mc}$) of its complementary graph: a chordal graph $G^c = (V, E^c)$. We first provide a general framework of the ``divide and conquer" heuristic of finding minimum biclique partitions of co-chordal graphs based on clique trees. Furthermore, a heuristic of complexity $O[|V|(|V|+|E^c|)]$ is proposed by applying lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of $G$ with size $\text{mc}(G^c)-1$. In addition, we prove that both of our heuristics can solve the minimum biclique partition problem on $G$ exactly if its complement $G^c$ is chordal and clique vertex irreducible. We also show that $\text{mc}(G^c) - 2 \leq \text{bp}(G) \leq \text{mc}(G^c) - 1$ if $G$ is a split graph

Constructing dense graphs with sublinear Hadwiger number

Mader asked to explicitly construct dense graphs for which the size of the largest clique minor is sublinear in the number of vertices. Such graphs exist as a random graph almost surely has this property. This question and variants were popularized by Thomason over several articles. We answer these questions by showing how to explicitly construct such graphs using blow-ups of small graphs with this property. This leads to the study of a fractional variant of the clique minor number, which may be of independent interest.Comment: 10 page

Efficient mining of discriminative molecular fragments

Frequent pattern discovery in structured data is receiving an increasing attention in many application areas of sciences. However, the computational complexity and the large amount of data to be explored often make the sequential algorithms unsuitable. In this context high performance distributed computing becomes a very interesting and promising approach. In this paper we present a parallel formulation of the frequent subgraph mining problem to discover interesting patterns in molecular compounds. The application is characterized by a highly irregular tree-structured computation. No estimation is available for task workloads, which show a power-law distribution in a wide range. The proposed approach allows dynamic resource aggregation and provides fault and latency tolerance. These features make the distributed application suitable for multi-domain heterogeneous environments, such as computational Grids. The distributed application has been evaluated on the well known National Cancer Institute’s HIV-screening dataset