On Bubble Generators in Directed Graphs

Bubbles are pairs of internally vertex-disjoint ( s , t )-paths in a directed graph, which have many applications in the processing of DNA and RNA data. Listing and analysing all bubbles in a given graph is usually unfeasible in practice, due to the exponential number of bubbles present in real data graphs. In this paper, we propose a notion of bubble generator set, i.e., a polynomial-sized subset of bubbles from which all the other bubbles can be obtained through a suitable application of a specific symmetric difference operator. This set provides a compact representation of the bubble space of a graph. A bubble generator can be useful in practice, since some pertinent information about all the bubbles can be more conveniently extracted from this compact set. We provide a polynomial-time algorithm to decompose any bubble of a graph into the bubbles of such a generator in a tree-like fashion. Finally, we present two applications of the bubble generator on a real RNA-seq dataset.