A mapping α:V(G)→V(H) from the vertex set of one graph G to another graph H is an isometric embedding if the shortest path distance between any two vertices in G equals the distance between their images in H. Here, we consider isometric embeddings of a weighted graph G into unweighted Hamming graphs, called Hamming embeddings, when G satisfies the property that every edge is a shortest path between its endpoints. Using a Cartesian product decomposition of G called its canonical isometric representation, we show that every Hamming embedding of G may be partitioned into a canonical partition, whose parts provide Hamming embeddings for each factor of the canonical isometric representation of G. This implies that G permits a Hamming embedding if and only if each factor of its canonical isometric representation is Hamming embeddable. This result extends prior work on unweighted graphs that showed that an unweighted graph permits a Hamming embedding if and only if each factor is a complete graph. When a graph G has nontrivial isometric representation, determining whether G has a Hamming embedding can be simplified to checking embeddability of two or more smaller graphs.

MOTIVATION: The study of T cell receptor (TCR) repertoires has generated new insights into immune system recognition. However, the ability to robustly characterize these populations has been limited by technical barriers and an inability to reliably infer heterodimeric chain pairings for TCRs. RESULTS: Here, we describe a novel analytical approach to an emerging immune repertoire sequencing method, improving the resolving power of this low-cost technology. This method relies upon the distribution of a T cell population across a 96-well plate, followed by barcoding and sequencing of the relevant transcripts from each T cell. Multicell Analytical Deconvolution for High Yield Paired-chain Evaluation (MAD-HYPE) uses Bayesian inference to more accurately extract TCR information, improving our ability to study and characterize T cell populations for immunology and immunotherapy applications. AVAILABILITY AND IMPLEMENTATION: The MAD-HYPE algorithm is released as an open-source project under the Apache License and is available from https://github.com/birnbaumlab/MAD-HYPE. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.