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Accurately genotyping structural variant (SV) alleles is crucial to genomics research. We present a novel method (kanpig) for genotyping SVs that leverages variant graphs and k-mer vectors to rapidly generate accurate SV genotypes. Benchmarking against the latest SV datasets shows kanpig achieves a single-sample genotyping concordance of 82.1%, significantly outperforming existing tools, which average 66.3%. We explore kanpig's use for multi-sample projects by testing on 47 genetically diverse samples and find kanpig accurately genotypes complex loci (e.g. SVs neighboring other SVs), and produces higher genotyping concordance than other tools. Kanpig requires only 43 seconds to process a single sample's 20x long-reads and can be run on PacBio or Oxford Nanopore long-reads.