We introduce a data structure, analysis, and visualization scheme called a cactus graph for comparing sets of related genomes. In common with multi-break point graphs and A-Bruijn graphs, cactus graphs can represent duplications and general genomic rearrangements, but additionally, they naturally decompose the common substructures in a set of related genomes into a hierarchy of chains that can be visualized as two-dimensional multiple alignments and nets that can be visualized in circular genome plots. Supplementary Material is available at www.liebertonline.com/cmb .