The bibliography of scientific papers lists items with variable degree of relevance for the contents of the paper itself. If we could identify the sources, i.e., the works that actually inspired the paper, their citations can help us uncover the genesis of scientific projects and would be more representative of the actual importance of papers and authors than the standard citation counts, when all references are considered. Here we present an analysis of the \textit{backbone of science}, i.e., the network of citations between papers and their sources. The latter are extracted from the full body of papers via Large Language Models (LLMs), which are currently very capable of correctly identifying the context in which a paper is cited. Using two different but related prompts, we find that the LLMs select only a small set of references, not taken at random, and that the resulting backbone networks are quite similar to each other with respect to their in-degree distributions, modularity, transitivity, and degree correlations. Backbone networks have higher heterogeneity in their in-degree distributions, compared to the full network, but the most cited papers are usually the same, with some important exceptions. Citation rankings among authors are also remarkably stable. We conclude that the full citation network, despite its redundancy with respect to the backbones, presents a reliable picture of the relative citation impact of papers and authors.

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