Bubbles are pairs of internally vertex-disjoint (s, t)-paths with applications in the processing of DNA and RNA data. For example, enumerating alternative splicing events in a reference-free context can be done by enumerating all bubbles in a de Bruijn graph built from RNA-seq reads [16]. However, 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 a bubble generator set, i.e. a polynomial-sized subset of bubbles from which all the others can be obtained through the application of a specific symmetric difference operator. This set provides a compact representation of the bubble space of a graph, which can be useful in practice since some pertinent information about all the bubbles can be more conveniently extracted from this compact set. Furthermore, 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.